Abstract group 36303120.b: $C_7^5:S_6.C_3$

• Label: 36303120.b
• Order: \( 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{5} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $42$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_7^5:S_6.C_3$
Order:	\(36303120\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{5} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_7^5:(C_6\times S_6)$, of order \(72606240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{5} \)
Composition factors:	$C_2$, $C_3$, $C_7$ x 5, $A_6$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	12	14	15	21	28	35	42
Elements	1	38955	351134	432180	345744	7707210	16806	6050520	1221570	4840416	3716160	2593080	2074464	6914880	36303120
Conjugacy classes	1	3	8	2	1	12	28	4	33	2	36	2	2	8	142
Divisions	1	3	5	2	1	7	27	2	29	1	20	2	1	3	104
Autjugacy classes	1	3	8	2	1	12	27	4	29	2	24	2	1	4	120

Dimension	1	2	5	9	10	16	18	20	32	36	60	90	108	120	144	180	216	240	270	360	480	540	720	960	1080	1440	2160
Irr. complex chars.	6	0	12	6	6	3	0	0	0	1	4	8	2	4	1	11	0	10	8	22	0	12	14	0	10	0	2	142
Irr. rational chars.	2	2	4	2	6	1	2	2	1	1	2	4	0	1	1	7	1	4	8	13	2	8	10	1	12	5	2	104

Minimal presentations

Permutation degree:	$42$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $42$ $\langle(1,14,15,25,3,13,21,23,7,11,20,22,5,10,18,26,6,12,17,27,2,9,19,24,4,8,16,28) \!\cdots\! \rangle$
Transitive group:	42T3176				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_7^5:S_6)$ . $C_3$	$(C_7^5:C_3)$ . $S_6$	$(C_7^5:A_6)$ . $C_6$	$(C_7^5:(C_3\times A_6))$ . $C_2$	more information

Elements of the group are displayed as permutations of degree 42.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_7^5:S_6.C_3$
Commutator:	$G' \simeq$ $C_7^5:A_6$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7^5:S_6.C_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^5$	$G/\operatorname{Fit} \simeq$ $C_3.A_6.C_2$
Radical:	$R \simeq$ $C_7^5:C_3$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_7^5$	$G/\operatorname{soc} \simeq$ $C_3.A_6.C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 4410 (order at least 8232) are shown, as well as all normal subgroups of any index.

Order 36303120: $C_7^5:S_6.C_3$

Order 18151560: $C_7^5:(C_3\times A_6)$

Order 12101040: $C_7^5:S_6$

Order 6050520: $C_7^5:A_6$, $C_7^5:(C_3\times S_5)$, $C_7^5:(C_3\times S_5)$

Order 3630312: $C_7^5:(S_3^2:C_6)$

Order 3025260: $C_7^5:\GL(2,4)$, $C_7^5:\GL(2,4)$

Order 2420208: $C_7^3.C_{14}^2.(C_6\times S_3)$, $C_7^5:(C_6\times S_4)$

Order 2016840: $C_7^5:S_5$, $C_7^5:S_5$

Order 1815156: $C_7^5.C_3^3.C_2^2$, $C_7^5:(C_3^2:C_{12})$, $C_7^5:(C_3\times S_3^2)$

Order 1210104: $C_7^5:(C_3\times S_4)$ x 2, $C_7^3.C_{14}^2.C_3.C_6$, $C_7^3.C_{14}^2.C_3.C_6$, $C_7^5.A_4.C_6$, $C_7^5:\SOPlus(4,2)$, $C_7^5:(C_6\times A_4)$

Order 1008420: $C_7^5.A_5$, $C_7\wr A_5$, $C_7\wr C_5:C_{12}$

Order 907578: $C_7^5.C_3^3.C_2$, $C_7^5.C_3^3.C_2$, $C_7^5:(S_3\times C_3^2)$

Order 864360: $C_7^4:(C_3\times S_5)$

Order 806736: $C_7^5.C_6.C_2^3$, $C_7^3.C_{14}^2.D_6$, $C_7^5:(C_2\times S_4)$

Order 605052: $C_7^5.(C_3\times A_4)$, $C_7^5.(C_6\times S_3)$, $C_7^5.C_3^2.C_4$, $C_7^5.S_3^2$, $C_7^5.S_3^2$, $C_7^5:(C_3\times A_4)$, $C_7^5:(C_6\times S_3)$

Order 518616: $C_7^4:S_3^2:C_6$

Order 504210: $C_7\wr C_5:C_6$

Order 453789: $C_7^5.C_3^3$

Order 432180: $C_7^4:\GL(2,4)$

Order 403368: $C_7^5.C_6.C_2^2$ x 2, $C_7^5.C_6.C_2^2$, $C_7^5.S_4$, $C_7^3.C_{14}^2.S_3$, $C_7^5.C_6.C_2^2$, $C_7^4.D_{28}:C_3$, $C_7^5.C_6.C_2^2$, $C_7^3.C_{14}^2.C_6$, $C_7^3.(C_7\times A_4).C_{14}$, $C_7^5.C_6.C_2^2$, $C_7^5.A_4.C_2$, $C_7^5:S_4$, $C_7^5:(C_2\times A_4)$, $C_7^5:(C_2\times A_4)$

Order 345744: $C_7^3.A_4:F_7.C_2$, $C_7^3.(A_4\times F_7).C_2$

Order 336140: $C_7^5.F_5$

Order 302526: $C_7^5:(C_3\times S_3)$ x 2, $C_7^5.C_3.C_6$, $C_7^4.C_3^2.C_{14}$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.(C_3\times S_3)$, $C_7^4.(S_3\times C_{21})$, $C_7^5:(C_3\times C_6)$

Order 288120: $C_7^4:S_5$

Order 268912: $C_7^5.C_2^3.C_2$

Order 259308: $C_7^4.C_3^3.C_2^2$, $C_7^4:(C_3\times S_3^2)$, $C_7^4:(C_3^2:C_{12})$

Order 252105: $C_7^5:C_{15}$

Order 201684: $C_7^5.C_6.C_2$ x 3, $C_7^5.C_{12}$ x 2, $C_7^5.C_6.C_2$, $C_7^5.A_4$, $C_7\times C_7^4:A_4$, $C_7^5.A_4$, $C_7^5.D_6$, $C_7^5.C_6.C_2$, $C_7^5:A_4$, $C_7^5:D_6$

Order 172872: $C_7^4:(C_3\times S_4)$ x 5, $C_7^4:(C_3\times S_4)$, $C_7^4:\SOPlus(4,2)$, $C_7^4:(C_3\times S_4)$, $C_7^4:(C_6\times A_4)$, $C_7^4:(C_6\times A_4)$, $C_7^4:(C_6\times A_4)$

Order 168070: $C_7^5:D_5$

Order 151263: $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$

Order 144060: $C_7^4:A_5$, $C_7^4:(C_3\times F_5)$

Order 134456: $C_7^5.D_4$ x 2, $C_7^5.C_2^3$, $C_7^4.C_{28}.C_2$, $C_7^4.(C_7\times D_4)$, $C_7^5.C_4.C_2$, $C_7^5.C_2^3$

Order 129654: $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^4:(C_3^2:C_6)$, $C_7^4:(S_3\times C_3^2)$

Order 115248: $C_7^4.C_6.C_2^3$ x 2, $C_7^4:(C_6\times D_4)$, $C_7^4:(C_2\times S_4)$, $D_7^3:D_{21}$

Order 100842: $C_7^5:C_6$ x 2, $C_7^5:S_3$ x 2, $C_7^5:S_3$, $C_7^5:S_3$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$

Order 86436: $C_7^4:(C_6\times S_3)$ x 8, $C_7^4:(C_3\times A_4)$ x 6, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.C_3^2.C_4$, $C_7^4:S_3^2$, $C_7^4:(C_6\times S_3)$

Order 84035: $C_7\wr C_5$

Order 72030: $C_7^4:(C_3\times D_5)$

Order 67228: $C_7^5:C_2^2$ x 3, $C_7^5:C_4$ x 2, $C_7^5:C_2^2$, $C_7^5:C_2^2$

Order 64827: $C_7^4:C_3^3$, $C_7^4:C_3^3$

Order 57624: $C_7\wr C_4:C_6$ x 10, $C_7^4:S_4$ x 5, $C_7\wr C_4:C_6$ x 4, $C_7^4.C_6.C_2^2$ x 3, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.(C_2\times A_4)$ x 2, $C_7^4.C_6.C_2^2$, $C_7^4:(C_2\times A_4)$, $C_7^4:(C_2\times A_4)$, $C_7\wr S_4$, $C_7^4:S_4$, $C_7^4:(C_2\times A_4)$, $D_7^3:C_{21}$, $C_7^3:A_4\times D_7$, $C_7^4:(C_2^2\times C_6)$, $C_7^4:(C_2\times C_{12})$

Order 50421: $C_7^5:C_3$, $C_7\times C_7^4:C_3$, $C_7^5:C_3$, $C_7^5:C_3$, $C_7^5:C_3$

Order 49392: $C_7^3:(C_6\times S_4)$, $C_7^3:(C_6\times S_4)$

Order 48020: $C_7^4:F_5$

Order 43218: $C_7^4:(C_3\times S_3)$ x 15, $C_7^4:(C_3\times S_3)$ x 8, $C_7^4:(C_3\times C_6)$ x 8, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^3:C_{21}:S_3$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$

Order 38416: $C_7^4.D_4.C_2$ x 2, $D_7^2:D_7^2$

Order 37044: $C_7^3.C_3^3.C_2^2$

Order 36015: $C_7^4:C_{15}$

Order 33614: $C_7^5:C_2$ x 2, $C_7^5:C_2$

Order 28812: $C_7^4:(C_2\times C_6)$ x 21, $C_7^4:(C_2\times C_6)$ x 13, $C_7^4:C_{12}$ x 10, $C_7^4:(C_2\times C_6)$ x 8, $C_7^3:D_{42}$ x 8, $C_7\wr A_4$ x 6, $C_7^4:A_4$ x 5, $C_7^4:(C_2\times C_6)$ x 4, $C_7^4:A_4$ x 2, $C_7^4:C_{12}$ x 2, $C_7^4:D_6$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$

Order 24696: $C_7^3:(C_3\times S_4)$ x 3, $C_7^3:(C_6\times A_4)$, $C_7^3:(C_3\times S_4)$, $D_7^3:C_3^2$

Order 24010: $C_7^4:D_5$

Order 21609: $C_7^4:C_3^2$ x 18, $C_7^4:C_3^2$ x 2, $C_7^4:C_3^2$ x 2, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$

Order 19208: $C_7^3:D_{28}$ x 10, $C_7\wr D_4$ x 4, $C_7^4:C_2^3$ x 3, $C_7^4:(C_2\times C_4)$ x 2, $C_7^4:D_4$ x 2, $C_7^4:D_4$ x 2, $C_7^4:C_2^3$ x 2, $C_7^4:C_2^3$, $C_7:D_7^3$, $C_7^4:(C_2\times C_4)$

Order 18522: $C_7^3:(C_3^2\times S_3)$, $C_7^3:(C_3\times C_3:S_3)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$

Order 16807: $C_7^5$

Order 16464: $D_7^3:C_6$ x 2, $C_7^3.C_6.C_2^3$, $C_7^3.C_6.C_2^3$, $C_2\times C_7^3:S_4$, $D_7\wr S_3$

Order 14406: $C_7^4:C_6$ x 37, $C_7^4:C_6$ x 28, $C_7^4:S_3$ x 15, $C_7^3:D_{21}$ x 8, $C_7^4:C_6$ x 8, $C_7^3:C_{42}$ x 8, $C_7^4:C_6$ x 6, $C_7^3:C_{42}$ x 2, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^3:C_{42}$, $C_7^4:C_6$, $C_7^3:C_{42}$, $C_7^3:C_{42}$, $C_7^4:C_6$, $C_7^3:C_{42}$

Order 12348: $C_7^3:(C_6\times S_3)$ x 8, $C_7^3:(C_3\times A_4)$ x 2, $C_7^3:S_3^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$

Order 12005: $C_7^4:C_5$

Order 9604: $C_7^4:C_2^2$ x 21, $C_7\wr C_2^2$ x 13, $C_7\wr C_4$ x 10, $C_7^4:C_2^2$ x 8, $C_7^4:C_2^2$ x 4, $C_7^4:C_4$ x 2, $C_7^4:C_2^2$, $C_7^4:C_2^2$

Order 9261: $C_7^3:C_3^3$ x 2, $C_7^3:C_3^3$

Order 8232: $D_7^2:F_7$ x 14, $D_7^3:C_3$ x 7, $C_7^3:(C_3\times D_4)$ x 2, $C_7^3:(C_2^2\times C_6)$ x 2, $(C_7\times D_{14}):F_7$ x 2, $C_2\times C_7^3:A_4$ x 2, $C_7^3:(C_2\times C_{12})$ x 2, $C_7^3:(C_3\times D_4)$ x 2, $(C_7\times D_7^2):C_6$ x 2, $C_7^3:S_4$ x 2, $D_7\wr C_3$ x 2, $C_7^3:(C_2\times C_{12})$, $C_7^3:(C_2^2\times C_6)$, $C_7^3:S_4$, $C_7^3:(C_2\times C_{12})$, $C_7^3:S_4$, $(C_7\times D_{14}):F_7$

Order 2160: $C_3\times S_6$

Order 27: $C_3^3$

Order 16: $C_2\times D_4$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 4410 (order at least 8232) are shown, as well as all normal subgroups of any index.

Order 36303120: $C_7^5:S_6.C_3$

Order 18151560: $C_7^5:(C_3\times A_6)$

Order 12101040: $C_7^5:S_6$

Order 6050520: $C_7^5:A_6$, $C_7^5:(C_3\times S_5)$, $C_7^5:(C_3\times S_5)$

Order 3630312: $C_7^5:(S_3^2:C_6)$

Order 3025260: $C_7^5:\GL(2,4)$, $C_7^5:\GL(2,4)$

Order 2420208: $C_7^3.C_{14}^2.(C_6\times S_3)$, $C_7^5:(C_6\times S_4)$

Order 2016840: $C_7^5:S_5$, $C_7^5:S_5$

Order 1815156: $C_7^5.C_3^3.C_2^2$, $C_7^5:(C_3^2:C_{12})$, $C_7^5:(C_3\times S_3^2)$

Order 1210104: $C_7^5:(C_3\times S_4)$ x 2, $C_7^3.C_{14}^2.C_3.C_6$, $C_7^3.C_{14}^2.C_3.C_6$, $C_7^5.A_4.C_6$, $C_7^5:\SOPlus(4,2)$, $C_7^5:(C_6\times A_4)$

Order 1008420: $C_7^5.A_5$, $C_7\wr A_5$, $C_7\wr C_5:C_{12}$

Order 907578: $C_7^5.C_3^3.C_2$, $C_7^5.C_3^3.C_2$, $C_7^5:(S_3\times C_3^2)$

Order 864360: $C_7^4:(C_3\times S_5)$

Order 806736: $C_7^5.C_6.C_2^3$, $C_7^3.C_{14}^2.D_6$, $C_7^5:(C_2\times S_4)$

Order 605052: $C_7^5.(C_3\times A_4)$, $C_7^5.(C_6\times S_3)$, $C_7^5.C_3^2.C_4$, $C_7^5.S_3^2$, $C_7^5.S_3^2$, $C_7^5:(C_3\times A_4)$, $C_7^5:(C_6\times S_3)$

Order 518616: $C_7^4:S_3^2:C_6$

Order 504210: $C_7\wr C_5:C_6$

Order 453789: $C_7^5.C_3^3$

Order 432180: $C_7^4:\GL(2,4)$

Order 403368: $C_7^5.C_6.C_2^2$ x 2, $C_7^5.C_6.C_2^2$, $C_7^5.S_4$, $C_7^3.C_{14}^2.S_3$, $C_7^5.C_6.C_2^2$, $C_7^4.D_{28}:C_3$, $C_7^5.C_6.C_2^2$, $C_7^3.C_{14}^2.C_6$, $C_7^3.(C_7\times A_4).C_{14}$, $C_7^5.C_6.C_2^2$, $C_7^5.A_4.C_2$, $C_7^5:S_4$, $C_7^5:(C_2\times A_4)$, $C_7^5:(C_2\times A_4)$

Order 345744: $C_7^3.A_4:F_7.C_2$, $C_7^3.(A_4\times F_7).C_2$

Order 336140: $C_7^5.F_5$

Order 302526: $C_7^5:(C_3\times S_3)$ x 2, $C_7^5.C_3.C_6$, $C_7^4.C_3^2.C_{14}$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.C_3.C_6$, $C_7^5.(C_3\times S_3)$, $C_7^4.(S_3\times C_{21})$, $C_7^5:(C_3\times C_6)$

Order 288120: $C_7^4:S_5$

Order 268912: $C_7^5.C_2^3.C_2$

Order 259308: $C_7^4.C_3^3.C_2^2$, $C_7^4:(C_3\times S_3^2)$, $C_7^4:(C_3^2:C_{12})$

Order 252105: $C_7^5:C_{15}$

Order 201684: $C_7^5.C_6.C_2$ x 3, $C_7^5.C_{12}$ x 2, $C_7^5.C_6.C_2$, $C_7^5.A_4$, $C_7\times C_7^4:A_4$, $C_7^5.A_4$, $C_7^5.D_6$, $C_7^5.C_6.C_2$, $C_7^5:A_4$, $C_7^5:D_6$

Order 172872: $C_7^4:(C_3\times S_4)$ x 5, $C_7^4:(C_3\times S_4)$, $C_7^4:\SOPlus(4,2)$, $C_7^4:(C_3\times S_4)$, $C_7^4:(C_6\times A_4)$, $C_7^4:(C_6\times A_4)$, $C_7^4:(C_6\times A_4)$

Order 168070: $C_7^5:D_5$

Order 151263: $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$, $C_7^5:C_3^2$

Order 144060: $C_7^4:A_5$, $C_7^4:(C_3\times F_5)$

Order 134456: $C_7^5.D_4$ x 2, $C_7^5.C_2^3$, $C_7^4.C_{28}.C_2$, $C_7^4.(C_7\times D_4)$, $C_7^5.C_4.C_2$, $C_7^5.C_2^3$

Order 129654: $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^4:(C_3^2:C_6)$, $C_7^4:(S_3\times C_3^2)$

Order 115248: $C_7^4.C_6.C_2^3$ x 2, $C_7^4:(C_6\times D_4)$, $C_7^4:(C_2\times S_4)$, $D_7^3:D_{21}$

Order 100842: $C_7^5:C_6$ x 2, $C_7^5:S_3$ x 2, $C_7^5:S_3$, $C_7^5:S_3$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$, $C_7^5:C_6$

Order 86436: $C_7^4:(C_6\times S_3)$ x 8, $C_7^4:(C_3\times A_4)$ x 6, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.C_3^2.C_4$, $C_7^4:S_3^2$, $C_7^4:(C_6\times S_3)$

Order 84035: $C_7\wr C_5$

Order 72030: $C_7^4:(C_3\times D_5)$

Order 67228: $C_7^5:C_2^2$ x 3, $C_7^5:C_4$ x 2, $C_7^5:C_2^2$, $C_7^5:C_2^2$

Order 64827: $C_7^4:C_3^3$, $C_7^4:C_3^3$

Order 57624: $C_7\wr C_4:C_6$ x 10, $C_7^4:S_4$ x 5, $C_7\wr C_4:C_6$ x 4, $C_7^4.C_6.C_2^2$ x 3, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.C_6.C_2^2$ x 2, $C_7^4.(C_2\times A_4)$ x 2, $C_7^4.C_6.C_2^2$, $C_7^4:(C_2\times A_4)$, $C_7^4:(C_2\times A_4)$, $C_7\wr S_4$, $C_7^4:S_4$, $C_7^4:(C_2\times A_4)$, $D_7^3:C_{21}$, $C_7^3:A_4\times D_7$, $C_7^4:(C_2^2\times C_6)$, $C_7^4:(C_2\times C_{12})$

Order 50421: $C_7^5:C_3$, $C_7\times C_7^4:C_3$, $C_7^5:C_3$, $C_7^5:C_3$, $C_7^5:C_3$

Order 49392: $C_7^3:(C_6\times S_4)$, $C_7^3:(C_6\times S_4)$

Order 48020: $C_7^4:F_5$

Order 43218: $C_7^4:(C_3\times S_3)$ x 15, $C_7^4:(C_3\times S_3)$ x 8, $C_7^4:(C_3\times C_6)$ x 8, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^3:C_{21}:S_3$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$

Order 38416: $C_7^4.D_4.C_2$ x 2, $D_7^2:D_7^2$

Order 37044: $C_7^3.C_3^3.C_2^2$

Order 36015: $C_7^4:C_{15}$

Order 33614: $C_7^5:C_2$ x 2, $C_7^5:C_2$

Order 28812: $C_7^4:(C_2\times C_6)$ x 21, $C_7^4:(C_2\times C_6)$ x 13, $C_7^4:C_{12}$ x 10, $C_7^4:(C_2\times C_6)$ x 8, $C_7^3:D_{42}$ x 8, $C_7\wr A_4$ x 6, $C_7^4:A_4$ x 5, $C_7^4:(C_2\times C_6)$ x 4, $C_7^4:A_4$ x 2, $C_7^4:C_{12}$ x 2, $C_7^4:D_6$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$

Order 24696: $C_7^3:(C_3\times S_4)$ x 3, $C_7^3:(C_6\times A_4)$, $C_7^3:(C_3\times S_4)$, $D_7^3:C_3^2$

Order 24010: $C_7^4:D_5$

Order 21609: $C_7^4:C_3^2$ x 18, $C_7^4:C_3^2$ x 2, $C_7^4:C_3^2$ x 2, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$

Order 19208: $C_7^3:D_{28}$ x 10, $C_7\wr D_4$ x 4, $C_7^4:C_2^3$ x 3, $C_7^4:(C_2\times C_4)$ x 2, $C_7^4:D_4$ x 2, $C_7^4:D_4$ x 2, $C_7^4:C_2^3$ x 2, $C_7^4:C_2^3$, $C_7:D_7^3$, $C_7^4:(C_2\times C_4)$

Order 18522: $C_7^3:(C_3^2\times S_3)$, $C_7^3:(C_3\times C_3:S_3)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$

Order 16807: $C_7^5$

Order 16464: $D_7^3:C_6$ x 2, $C_7^3.C_6.C_2^3$, $C_7^3.C_6.C_2^3$, $C_2\times C_7^3:S_4$, $D_7\wr S_3$

Order 14406: $C_7^4:C_6$ x 37, $C_7^4:C_6$ x 28, $C_7^4:S_3$ x 15, $C_7^3:D_{21}$ x 8, $C_7^4:C_6$ x 8, $C_7^3:C_{42}$ x 8, $C_7^4:C_6$ x 6, $C_7^3:C_{42}$ x 2, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7^3:C_{42}$, $C_7^4:C_6$, $C_7^3:C_{42}$, $C_7^3:C_{42}$, $C_7^4:C_6$, $C_7^3:C_{42}$

Order 12348: $C_7^3:(C_6\times S_3)$ x 8, $C_7^3:(C_3\times A_4)$ x 2, $C_7^3:S_3^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$

Order 12005: $C_7^4:C_5$

Order 9604: $C_7^4:C_2^2$ x 21, $C_7\wr C_2^2$ x 13, $C_7\wr C_4$ x 10, $C_7^4:C_2^2$ x 8, $C_7^4:C_2^2$ x 4, $C_7^4:C_4$ x 2, $C_7^4:C_2^2$, $C_7^4:C_2^2$

Order 9261: $C_7^3:C_3^3$ x 2, $C_7^3:C_3^3$

Order 8232: $D_7^2:F_7$ x 14, $D_7^3:C_3$ x 7, $C_7^3:(C_3\times D_4)$ x 2, $C_7^3:(C_2^2\times C_6)$ x 2, $(C_7\times D_{14}):F_7$ x 2, $C_2\times C_7^3:A_4$ x 2, $C_7^3:(C_2\times C_{12})$ x 2, $C_7^3:(C_3\times D_4)$ x 2, $(C_7\times D_7^2):C_6$ x 2, $C_7^3:S_4$ x 2, $D_7\wr C_3$ x 2, $C_7^3:(C_2\times C_{12})$, $C_7^3:(C_2^2\times C_6)$, $C_7^3:S_4$, $C_7^3:(C_2\times C_{12})$, $C_7^3:S_4$, $(C_7\times D_{14}):F_7$

Order 2160: $C_3\times S_6$

Order 27: $C_3^3$

Order 16: $C_2\times D_4$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 36303120: $C_7^5:S_6.C_3$ ($C_1$)

Order 18151560: $C_7^5:(C_3\times A_6)$ ($C_2$)

Order 12101040: $C_7^5:S_6$ ($C_3$)

Order 6050520: $C_7^5:A_6$ ($C_6$)

Order 50421: $C_7^5:C_3$ ($S_6$)

Order 16807: $C_7^5$ ($C_3.A_6.C_2$)

Order 1: $C_1$ ($C_7^5:S_6.C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 36303120: $C_7^5:S_6.C_3$ ($C_1$)

Order 18151560: $C_7^5:(C_3\times A_6)$ ($C_2$)

Order 12101040: $C_7^5:S_6$ ($C_3$)

Order 6050520: $C_7^5:A_6$ ($C_6$)

Order 50421: $C_7^5:C_3$ ($S_6$)

Order 16807: $C_7^5$ ($C_3.A_6.C_2$)

Order 1: $C_1$ ($C_7^5:S_6.C_3$)

Series

Derived series	$C_7^5:S_6.C_3$	$\rhd$	$C_7^5:A_6$
Chief series	$C_7^5:S_6.C_3$	$\rhd$	$C_7^5:(C_3\times A_6)$	$\rhd$	$C_7^5:C_3$	$\rhd$	$C_7^5$	$\rhd$	$C_1$
Lower central series	$C_7^5:S_6.C_3$	$\rhd$	$C_7^5:A_6$
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 1 larger groups in the database.

Character theory

Complex character table

See the $142 \times 142$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $104 \times 104$ rational character table.