Abstract group 5040.bh: $S_3\times F_5\times F_7$

• Label: 5040.bh
• Order: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $15$
• Trans deg.: $105$
• Rank: $3$

Group information

Description:	$S_3\times F_5\times F_7$
Order:	\(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_3\times F_5\times F_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	21	28	30	35	42	70	84	105
Elements	1	191	44	320	4	1060	6	124	1840	138	176	12	240	560	24	60	72	120	48	5040
Conjugacy classes	1	7	5	8	1	23	1	3	28	3	5	1	4	9	1	1	1	2	1	105
Divisions	1	7	3	4	1	13	1	3	8	3	3	1	2	5	1	1	1	1	1	60
Autjugacy classes	1	7	5	8	1	23	1	3	28	3	5	1	4	9	1	1	1	2	1	105

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	48	24	12	8	6	4	0	2	1	105
Irr. rational chars.	8	16	14	4	8	4	2	3	1	60

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$105$
Rank:	$3$
Inequivalent generating triples:	$6709248$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	48	48
Arbitrary	12	12	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{12}=d^{105}=[a,b]=[b,c]=1, c^{a}=cd^{35}, d^{a}=d^{71}, d^{b}=d^{34}, d^{c}=d^{68} \rangle$
Permutation group:	Degree $15$ $\langle(8,14,12,9)(11,15), (8,14,10,9,12)(11,13,15), (1,4)(2,6)(5,7), (11,13), (1,5,2)(4,7,6), (1,4,5,2,3,6,7), (11,13,15), (8,12)(9,14)\rangle$
Direct product:	$S_3$ $\, \times\, $ $F_5$ $\, \times\, $ $F_7$
Semidirect product:	$(D_{15}\times F_7)$ $\,\rtimes\,$ $C_4$	$(D_{15}:F_7)$ $\,\rtimes\,$ $C_4$	$(F_5\times D_{21})$ $\,\rtimes\,$ $C_6$	$D_{21}$ $\,\rtimes\,$ $(C_6\times F_5)$	all 73
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5\times F_7)$ . $D_6$	$D_5$ . $(D_6\times F_7)$	$(D_{15}:D_{14})$ . $C_6$	$(S_3\times D_5\times F_7)$ . $C_2$	all 17
Aut. group:	$\Aut(D_{105})$	$\Aut(C_{105}:C_4)$	$\Aut(D_7\times D_{15})$	$\Aut(D_5\times D_{21})$	all 33

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 7656 subgroups in 464 conjugacy classes, 98 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times F_5\times F_7$
Commutator:	$G' \simeq$ $C_{105}$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times F_5\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_{105}$	$G/\operatorname{Fit} \simeq$ $C_2^2\times C_{12}$
Radical:	$R \simeq$ $S_3\times F_5\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{105}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 5040: $S_3\times F_5\times F_7$

Order 2520: $C_{15}:C_4\times F_7$, $D_{105}:C_{12}$, $C_3\times F_5\times F_7$, $S_3\times C_{35}:C_{12}$, $D_{105}:C_{12}$, $S_3\times C_{35}:C_{12}$, $S_3\times D_5\times F_7$

Order 1680: $S_3\times D_7\times F_5$, $D_{70}:C_{12}$

Order 1260: $D_{15}:F_7$, $D_{105}:C_6$, $D_{105}:C_6$, $C_{35}:C_6\times S_3$, $D_{15}\times F_7$, $D_{21}:C_{30}$, $C_{35}:C_6^2$, $C_{105}:C_{12}$, $C_{105}:C_{12}$, $C_{105}:C_{12}$, $C_{105}:C_{12}$

Order 1008: $C_4\times S_3\times F_7$

Order 840: $F_5\times F_7$ x 5, $D_{15}:D_{14}$, $D_{105}:C_4$, $C_{35}:C_4\times S_3$, $F_5\times D_{21}$, $D_{15}:C_{28}$, $C_{15}:C_4\times D_7$, $D_{35}:C_{12}$, $D_{10}\times F_7$, $C_{70}:C_{12}$, $C_{70}:C_{12}$

Order 720: $D_{30}:C_{12}$

Order 630: $C_{105}:C_6$, $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{21}:C_{30}$, $C_{105}:C_6$, $C_{105}:C_6$, $C_{15}\times F_7$

Order 560: $F_5\times D_{14}$

Order 504: $C_{84}:C_6$, $C_{84}:C_6$, $C_{12}\times F_7$, $D_{21}:C_{12}$, $D_6.F_7$, $C_3:C_4\times F_7$, $D_6\times F_7$

Order 420: $D_5\times F_7$ x 5, $C_{35}:C_{12}$ x 3, $C_{35}:C_{12}$ x 3, $C_{15}:D_{14}$, $S_3\times D_{35}$, $D_5\times D_{21}$, $C_{21}:D_{10}$, $D_7\times D_{15}$, $C_{15}:D_{14}$, $C_{15}:D_{14}$, $C_{105}:C_4$, $C_{15}:C_{28}$, $C_{105}:C_4$, $F_5\times C_{21}$, $C_{70}:C_6$, $C_{70}:C_6$, $C_{10}\times F_7$

Order 360: $D_{15}:C_{12}$ x 4, $C_{30}:D_6$, $C_{30}:C_{12}$, $C_{30}:C_{12}$

Order 336: $C_{12}:D_{14}$, $D_{14}:C_{12}$

Order 315: $C_{105}:C_3$

Order 280: $D_7\times F_5$ x 4, $D_5\times D_{14}$, $C_{70}:C_4$, $C_{14}\times F_5$

Order 252: $S_3\times F_7$ x 4, $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $D_{10}:C_{12}$, $D_6\times F_5$

Order 210: $C_{35}:C_6$ x 3, $C_{35}:C_6$ x 3, $C_5\times F_7$ x 3, $C_5\times D_{21}$, $S_3\times C_{35}$, $C_3\times D_{35}$, $D_5\times C_{21}$, $D_7\times C_{15}$, $C_7:C_{30}$, $D_{105}$, $C_7\times D_{15}$

Order 180: $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 2, $C_3^2\times F_5$ x 2, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$

Order 168: $C_4\times F_7$ x 5, $S_3\times D_{14}$, $C_2^2\times F_7$, $C_4\times D_{21}$, $S_3\times C_{28}$, $C_{12}\times D_7$, $C_{14}:C_{12}$, $D_{21}:C_4$, $C_6.D_{14}$, $C_6.D_{14}$, $C_{14}:C_{12}$

Order 144: $C_{12}\times D_6$

Order 140: $D_5\times D_7$ x 4, $C_{35}:C_4$ x 2, $C_7\times F_5$ x 2, $C_7\times D_{10}$, $C_5\times D_{14}$, $D_{70}$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$ x 2, $C_3\times F_7$ x 2, $C_{21}:C_6$

Order 120: $C_6\times F_5$ x 8, $S_3\times F_5$ x 4, $C_6\times D_{10}$, $S_3\times D_{10}$, $C_{30}:C_4$

Order 112: $C_4\times D_{14}$

Order 105: $C_7:C_{15}$ x 2, $C_{105}$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 84: $C_2\times F_7$ x 7, $S_3\times D_7$ x 4, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3, $C_{14}:C_6$, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 80: $C_2^2\times F_5$

Order 72: $S_3\times C_{12}$ x 4, $C_6\times D_6$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 70: $D_{35}$ x 2, $C_5\times D_7$ x 2, $C_7\times D_5$ x 2, $C_{70}$

Order 63: $C_{21}:C_3$

Order 60: $C_3\times F_5$ x 8, $C_3\times D_{10}$ x 8, $S_3\times D_5$ x 4, $C_{15}:C_4$ x 2, $C_2\times C_{30}$, $D_{30}$, $S_3\times C_{10}$

Order 56: $C_4\times D_7$ x 4, $C_2\times C_{28}$, $C_{14}:C_4$, $C_2\times D_{14}$

Order 48: $C_2^2\times C_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 42: $F_7$ x 6, $C_7:C_6$ x 4, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 40: $C_2\times F_5$ x 6, $C_2\times D_{10}$

Order 36: $C_6\times S_3$ x 6, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6^2$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 5, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 28: $D_{14}$ x 6, $C_{28}$ x 2, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 8, $C_4\times S_3$ x 4, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 6, $F_5$ x 4, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$ x 3

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 8, $D_6$ x 6, $C_3:C_4$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 13, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5040: $S_3\times F_5\times F_7$

Order 2520: $C_{15}:C_4\times F_7$, $D_{105}:C_{12}$, $C_3\times F_5\times F_7$, $S_3\times C_{35}:C_{12}$, $D_{105}:C_{12}$, $S_3\times C_{35}:C_{12}$, $S_3\times D_5\times F_7$

Order 1680: $S_3\times D_7\times F_5$, $D_{70}:C_{12}$

Order 1260: $D_{15}:F_7$, $D_{105}:C_6$, $D_{105}:C_6$, $C_{35}:C_6\times S_3$, $D_{15}\times F_7$, $D_{21}:C_{30}$, $C_{35}:C_6^2$, $C_{105}:C_{12}$, $C_{105}:C_{12}$, $C_{105}:C_{12}$, $C_{105}:C_{12}$

Order 1008: $C_4\times S_3\times F_7$

Order 840: $F_5\times F_7$ x 5, $D_{15}:D_{14}$, $D_{105}:C_4$, $C_{35}:C_4\times S_3$, $F_5\times D_{21}$, $D_{15}:C_{28}$, $C_{15}:C_4\times D_7$, $D_{35}:C_{12}$, $D_{10}\times F_7$, $C_{70}:C_{12}$, $C_{70}:C_{12}$

Order 720: $D_{30}:C_{12}$

Order 630: $C_{105}:C_6$, $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{21}:C_{30}$, $C_{105}:C_6$, $C_{105}:C_6$, $C_{15}\times F_7$

Order 560: $F_5\times D_{14}$

Order 504: $C_{84}:C_6$, $C_{84}:C_6$, $C_{12}\times F_7$, $D_{21}:C_{12}$, $D_6.F_7$, $C_3:C_4\times F_7$, $D_6\times F_7$

Order 420: $D_5\times F_7$ x 5, $C_{35}:C_{12}$ x 3, $C_{35}:C_{12}$ x 3, $C_{15}:D_{14}$, $S_3\times D_{35}$, $D_5\times D_{21}$, $C_{21}:D_{10}$, $D_7\times D_{15}$, $C_{15}:D_{14}$, $C_{15}:D_{14}$, $C_{105}:C_4$, $C_{15}:C_{28}$, $C_{105}:C_4$, $F_5\times C_{21}$, $C_{70}:C_6$, $C_{70}:C_6$, $C_{10}\times F_7$

Order 360: $D_{15}:C_{12}$ x 4, $C_{30}:D_6$, $C_{30}:C_{12}$, $C_{30}:C_{12}$

Order 336: $C_{12}:D_{14}$, $D_{14}:C_{12}$

Order 315: $C_{105}:C_3$

Order 280: $D_7\times F_5$ x 4, $D_5\times D_{14}$, $C_{70}:C_4$, $C_{14}\times F_5$

Order 252: $S_3\times F_7$ x 4, $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $D_{10}:C_{12}$, $D_6\times F_5$

Order 210: $C_{35}:C_6$ x 3, $C_{35}:C_6$ x 3, $C_5\times F_7$ x 3, $C_5\times D_{21}$, $S_3\times C_{35}$, $C_3\times D_{35}$, $D_5\times C_{21}$, $D_7\times C_{15}$, $C_7:C_{30}$, $D_{105}$, $C_7\times D_{15}$

Order 180: $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 2, $C_3^2\times F_5$ x 2, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$

Order 168: $C_4\times F_7$ x 5, $S_3\times D_{14}$, $C_2^2\times F_7$, $C_4\times D_{21}$, $S_3\times C_{28}$, $C_{12}\times D_7$, $C_{14}:C_{12}$, $D_{21}:C_4$, $C_6.D_{14}$, $C_6.D_{14}$, $C_{14}:C_{12}$

Order 144: $C_{12}\times D_6$

Order 140: $D_5\times D_7$ x 4, $C_{35}:C_4$ x 2, $C_7\times F_5$ x 2, $C_7\times D_{10}$, $C_5\times D_{14}$, $D_{70}$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$ x 2, $C_3\times F_7$ x 2, $C_{21}:C_6$

Order 120: $C_6\times F_5$ x 8, $S_3\times F_5$ x 4, $C_6\times D_{10}$, $S_3\times D_{10}$, $C_{30}:C_4$

Order 112: $C_4\times D_{14}$

Order 105: $C_7:C_{15}$ x 2, $C_{105}$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 84: $C_2\times F_7$ x 7, $S_3\times D_7$ x 4, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3, $C_{14}:C_6$, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 80: $C_2^2\times F_5$

Order 72: $S_3\times C_{12}$ x 4, $C_6\times D_6$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 70: $D_{35}$ x 2, $C_5\times D_7$ x 2, $C_7\times D_5$ x 2, $C_{70}$

Order 63: $C_{21}:C_3$

Order 60: $C_3\times F_5$ x 8, $C_3\times D_{10}$ x 8, $S_3\times D_5$ x 4, $C_{15}:C_4$ x 2, $C_2\times C_{30}$, $D_{30}$, $S_3\times C_{10}$

Order 56: $C_4\times D_7$ x 4, $C_2\times C_{28}$, $C_{14}:C_4$, $C_2\times D_{14}$

Order 48: $C_2^2\times C_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 42: $F_7$ x 6, $C_7:C_6$ x 4, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 40: $C_2\times F_5$ x 6, $C_2\times D_{10}$

Order 36: $C_6\times S_3$ x 6, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6^2$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 5, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 28: $D_{14}$ x 6, $C_{28}$ x 2, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 8, $C_4\times S_3$ x 4, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 6, $F_5$ x 4, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$ x 3

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 8, $D_6$ x 6, $C_3:C_4$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 13, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5040: $S_3\times F_5\times F_7$ ($C_1$)

Order 2520: $C_{15}:C_4\times F_7$ ($C_2$), $D_{105}:C_{12}$ ($C_2$), $C_3\times F_5\times F_7$ ($C_2$), $S_3\times C_{35}:C_{12}$ ($C_2$), $D_{105}:C_{12}$ ($C_2$), $S_3\times C_{35}:C_{12}$ ($C_2$), $S_3\times D_5\times F_7$ ($C_2$)

Order 1680: $S_3\times D_7\times F_5$ ($C_3$)

Order 1260: $D_{15}:F_7$ ($C_4$), $D_{105}:C_6$ ($C_4$), $D_{105}:C_6$ ($C_2^2$), $C_{35}:C_6\times S_3$ ($C_2^2$), $D_{15}\times F_7$ ($C_4$), $D_{21}:C_{30}$ ($C_4$), $C_{35}:C_6^2$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$)

Order 840: $D_{15}:D_{14}$ ($C_6$), $D_{105}:C_4$ ($C_6$), $C_{35}:C_4\times S_3$ ($C_6$), $F_5\times D_{21}$ ($C_6$), $D_{15}:C_{28}$ ($C_6$), $C_{15}:C_4\times D_7$ ($C_6$), $D_{35}:C_{12}$ ($C_6$), $F_5\times F_7$ ($S_3$)

Order 630: $C_{105}:C_6$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2\times C_4$), $C_{21}:C_{30}$ ($C_2\times C_4$), $C_{21}:C_{30}$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2^3$), $C_{15}\times F_7$ ($C_2\times C_4$)

Order 420: $C_{15}:D_{14}$ ($C_{12}$), $S_3\times D_{35}$ ($C_{12}$), $D_5\times D_{21}$ ($C_2\times C_6$), $C_{21}:D_{10}$ ($C_2\times C_6$), $D_7\times D_{15}$ ($C_{12}$), $C_{15}:D_{14}$ ($C_{12}$), $C_{15}:D_{14}$ ($C_2\times C_6$), $C_{105}:C_4$ ($C_2\times C_6$), $C_{15}:C_{28}$ ($C_2\times C_6$), $C_{105}:C_4$ ($C_2\times C_6$), $F_5\times C_{21}$ ($C_2\times C_6$), $D_5\times F_7$ ($D_6$), $C_{35}:C_{12}$ ($D_6$), $C_{35}:C_{12}$ ($D_6$)

Order 315: $C_{105}:C_3$ ($C_2^2\times C_4$)

Order 280: $D_7\times F_5$ ($C_3\times S_3$)

Order 252: $S_3\times F_7$ ($F_5$)

Order 210: $C_5\times D_{21}$ ($C_2\times C_{12}$), $S_3\times C_{35}$ ($C_2\times C_{12}$), $C_3\times D_{35}$ ($C_2\times C_{12}$), $D_5\times C_{21}$ ($C_2^2\times C_6$), $D_7\times C_{15}$ ($C_2\times C_{12}$), $C_{35}:C_6$ ($C_4\times S_3$), $C_{35}:C_6$ ($C_2\times D_6$), $D_{105}$ ($C_2\times C_{12}$), $C_7\times D_{15}$ ($C_2\times C_{12}$), $C_5\times F_7$ ($C_4\times S_3$)

Order 140: $D_5\times D_7$ ($C_6\times S_3$), $C_{35}:C_4$ ($C_6\times S_3$), $C_7\times F_5$ ($C_6\times S_3$)

Order 126: $C_{21}:C_6$ ($C_2\times F_5$), $C_{21}:C_6$ ($C_2\times F_5$), $C_3\times F_7$ ($C_2\times F_5$)

Order 120: $S_3\times F_5$ ($F_7$)

Order 105: $C_{105}$ ($C_2^2\times C_{12}$), $C_7:C_{15}$ ($C_4\times D_6$)

Order 84: $S_3\times D_7$ ($C_3\times F_5$)

Order 70: $D_{35}$ ($S_3\times C_{12}$), $C_5\times D_7$ ($S_3\times C_{12}$), $C_7\times D_5$ ($C_6\times D_6$)

Order 63: $C_{21}:C_3$ ($C_2^2\times F_5$)

Order 60: $S_3\times D_5$ ($C_2\times F_7$), $C_{15}:C_4$ ($C_2\times F_7$), $C_3\times F_5$ ($C_2\times F_7$)

Order 42: $D_{21}$ ($C_6\times F_5$), $C_3\times D_7$ ($C_6\times F_5$), $S_3\times C_7$ ($C_6\times F_5$), $F_7$ ($S_3\times F_5$)

Order 35: $C_{35}$ ($C_{12}\times D_6$)

Order 30: $D_{15}$ ($C_4\times F_7$), $C_3\times D_5$ ($C_2^2\times F_7$), $C_5\times S_3$ ($C_4\times F_7$)

Order 21: $C_{21}$ ($D_{10}:C_{12}$), $C_7:C_3$ ($D_6\times F_5$)

Order 20: $F_5$ ($S_3\times F_7$)

Order 15: $C_{15}$ ($D_{14}:C_{12}$)

Order 14: $D_7$ ($D_{15}:C_{12}$)

Order 10: $D_5$ ($D_6\times F_7$)

Order 7: $C_7$ ($D_{30}:C_{12}$)

Order 6: $S_3$ ($F_5\times F_7$)

Order 5: $C_5$ ($C_4\times S_3\times F_7$)

Order 3: $C_3$ ($D_{70}:C_{12}$)

Order 1: $C_1$ ($S_3\times F_5\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5040: $S_3\times F_5\times F_7$ ($C_1$)

Order 2520: $C_{15}:C_4\times F_7$ ($C_2$), $D_{105}:C_{12}$ ($C_2$), $C_3\times F_5\times F_7$ ($C_2$), $S_3\times C_{35}:C_{12}$ ($C_2$), $D_{105}:C_{12}$ ($C_2$), $S_3\times C_{35}:C_{12}$ ($C_2$), $S_3\times D_5\times F_7$ ($C_2$)

Order 1680: $S_3\times D_7\times F_5$ ($C_3$)

Order 1260: $D_{15}:F_7$ ($C_4$), $D_{105}:C_6$ ($C_4$), $D_{105}:C_6$ ($C_2^2$), $C_{35}:C_6\times S_3$ ($C_2^2$), $D_{15}\times F_7$ ($C_4$), $D_{21}:C_{30}$ ($C_4$), $C_{35}:C_6^2$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$), $C_{105}:C_{12}$ ($C_2^2$)

Order 840: $D_{15}:D_{14}$ ($C_6$), $D_{105}:C_4$ ($C_6$), $C_{35}:C_4\times S_3$ ($C_6$), $F_5\times D_{21}$ ($C_6$), $D_{15}:C_{28}$ ($C_6$), $C_{15}:C_4\times D_7$ ($C_6$), $D_{35}:C_{12}$ ($C_6$), $F_5\times F_7$ ($S_3$)

Order 630: $C_{105}:C_6$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2\times C_4$), $C_{21}:C_{30}$ ($C_2\times C_4$), $C_{21}:C_{30}$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2\times C_4$), $C_{105}:C_6$ ($C_2^3$), $C_{15}\times F_7$ ($C_2\times C_4$)

Order 420: $C_{15}:D_{14}$ ($C_{12}$), $S_3\times D_{35}$ ($C_{12}$), $D_5\times D_{21}$ ($C_2\times C_6$), $C_{21}:D_{10}$ ($C_2\times C_6$), $D_7\times D_{15}$ ($C_{12}$), $C_{15}:D_{14}$ ($C_{12}$), $C_{15}:D_{14}$ ($C_2\times C_6$), $C_{105}:C_4$ ($C_2\times C_6$), $C_{15}:C_{28}$ ($C_2\times C_6$), $C_{105}:C_4$ ($C_2\times C_6$), $F_5\times C_{21}$ ($C_2\times C_6$), $D_5\times F_7$ ($D_6$), $C_{35}:C_{12}$ ($D_6$), $C_{35}:C_{12}$ ($D_6$)

Order 315: $C_{105}:C_3$ ($C_2^2\times C_4$)

Order 280: $D_7\times F_5$ ($C_3\times S_3$)

Order 252: $S_3\times F_7$ ($F_5$)

Order 210: $C_5\times D_{21}$ ($C_2\times C_{12}$), $S_3\times C_{35}$ ($C_2\times C_{12}$), $C_3\times D_{35}$ ($C_2\times C_{12}$), $D_5\times C_{21}$ ($C_2^2\times C_6$), $D_7\times C_{15}$ ($C_2\times C_{12}$), $C_{35}:C_6$ ($C_4\times S_3$), $C_{35}:C_6$ ($C_2\times D_6$), $D_{105}$ ($C_2\times C_{12}$), $C_7\times D_{15}$ ($C_2\times C_{12}$), $C_5\times F_7$ ($C_4\times S_3$)

Order 140: $D_5\times D_7$ ($C_6\times S_3$), $C_{35}:C_4$ ($C_6\times S_3$), $C_7\times F_5$ ($C_6\times S_3$)

Order 126: $C_{21}:C_6$ ($C_2\times F_5$), $C_{21}:C_6$ ($C_2\times F_5$), $C_3\times F_7$ ($C_2\times F_5$)

Order 120: $S_3\times F_5$ ($F_7$)

Order 105: $C_{105}$ ($C_2^2\times C_{12}$), $C_7:C_{15}$ ($C_4\times D_6$)

Order 84: $S_3\times D_7$ ($C_3\times F_5$)

Order 70: $D_{35}$ ($S_3\times C_{12}$), $C_5\times D_7$ ($S_3\times C_{12}$), $C_7\times D_5$ ($C_6\times D_6$)

Order 63: $C_{21}:C_3$ ($C_2^2\times F_5$)

Order 60: $S_3\times D_5$ ($C_2\times F_7$), $C_{15}:C_4$ ($C_2\times F_7$), $C_3\times F_5$ ($C_2\times F_7$)

Order 42: $D_{21}$ ($C_6\times F_5$), $C_3\times D_7$ ($C_6\times F_5$), $S_3\times C_7$ ($C_6\times F_5$), $F_7$ ($S_3\times F_5$)

Order 35: $C_{35}$ ($C_{12}\times D_6$)

Order 30: $D_{15}$ ($C_4\times F_7$), $C_3\times D_5$ ($C_2^2\times F_7$), $C_5\times S_3$ ($C_4\times F_7$)

Order 21: $C_{21}$ ($D_{10}:C_{12}$), $C_7:C_3$ ($D_6\times F_5$)

Order 20: $F_5$ ($S_3\times F_7$)

Order 15: $C_{15}$ ($D_{14}:C_{12}$)

Order 14: $D_7$ ($D_{15}:C_{12}$)

Order 10: $D_5$ ($D_6\times F_7$)

Order 7: $C_7$ ($D_{30}:C_{12}$)

Order 6: $S_3$ ($F_5\times F_7$)

Order 5: $C_5$ ($C_4\times S_3\times F_7$)

Order 3: $C_3$ ($D_{70}:C_{12}$)

Order 1: $C_1$ ($S_3\times F_5\times F_7$)

Series

Derived series

$S_3\times F_5\times F_7$

$\rhd$

$C_{105}$

$\rhd$

$C_1$

Chief series

$S_3\times F_5\times F_7$

$\rhd$

$S_3\times D_5\times F_7$

$\rhd$

$C_{35}:C_6\times S_3$

$\rhd$

$C_{105}:C_6$

$\rhd$

$C_{105}:C_3$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$S_3\times F_5\times F_7$

$\rhd$

$C_{105}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.