Abstract group 1260.71: $D_{105}:C_6$

• Label: 1260.71
• Order: \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $105$
• Rank: $2$

Group information

Description:	$D_{105}:C_6$
Order:	\(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \)
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 2, $C_3$ 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	5	6	7	10	14	15	21	30	35	42	105
Elements	1	131	44	4	472	6	84	30	176	12	168	24	60	48	1260
Conjugacy classes	1	3	5	2	9	1	2	1	10	2	4	2	2	4	48
Divisions	1	3	3	1	5	1	1	1	3	1	1	1	1	1	24
Autjugacy classes	1	3	5	1	9	1	1	1	5	1	2	1	1	1	33

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	12	18	6	6	0	6	0	0	0	48
Irr. rational chars.	4	6	4	2	3	2	1	1	1	24

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$105$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	48
Arbitrary	8	8	12

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{105}=[a,b]=1, c^{a}=c^{64}, c^{b}=c^{26} \rangle$
Permutation group:	Degree $15$ $\langle(7,8)(10,11)(12,13)(14,15), (2,3)(4,5), (10,12,14)(11,13,15), (6,7,8), (1,2,4,5,3), (9,10,12,15,14,13,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 69 & 20 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 36 & 35 \\ 35 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/70\Z)$
Direct product:	$D_5$ $\, \times\, $ $(C_{21}:C_6)$
Semidirect product:	$D_{105}$ $\,\rtimes\,$ $C_6$	$(C_3\times D_5)$ $\,\rtimes\,$ $F_7$	$C_3$ $\,\rtimes\,$ $(D_5\times F_7)$	$(C_{35}:C_6)$ $\,\rtimes\,$ $S_3$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 1184 subgroups in 88 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{105}:C_6$
Commutator:	$G' \simeq$ $C_{105}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{105}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{105}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_6$
Radical:	$R \simeq$ $D_{105}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{105}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 1260: $D_{105}:C_6$

Order 630: $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{105}:C_6$

Order 420: $D_5\times D_{21}$, $D_5\times F_7$

Order 315: $C_{105}:C_3$

Order 252: $C_{42}:C_6$

Order 210: $C_{35}:C_6$ x 2, $C_5\times D_{21}$, $D_5\times C_{21}$, $C_{35}:C_6$, $D_{105}$, $C_5\times F_7$

Order 180: $C_{15}:D_6$

Order 140: $D_5\times D_7$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$

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

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

Order 84: $C_2\times F_7$, $D_{42}$

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

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$, $C_3\times D_{10}$

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

Order 42: $D_{21}$ x 2, $C_7:C_6$ x 2, $F_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 4, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 28: $D_{14}$

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

Order 20: $D_{10}$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 15: $C_{15}$ x 3

Order 14: $D_7$ x 2, $C_{14}$

Order 12: $C_2\times C_6$, $D_6$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1260: $D_{105}:C_6$

Order 630: $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{105}:C_6$

Order 420: $D_5\times D_{21}$, $D_5\times F_7$

Order 315: $C_{105}:C_3$

Order 252: $C_{42}:C_6$

Order 210: $C_{35}:C_6$ x 2, $C_5\times D_{21}$, $D_5\times C_{21}$, $C_{35}:C_6$, $D_{105}$, $C_5\times F_7$

Order 180: $C_{15}:D_6$

Order 140: $D_5\times D_7$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$

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

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

Order 84: $C_2\times F_7$, $D_{42}$

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

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$, $C_3\times D_{10}$

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

Order 42: $D_{21}$ x 2, $C_7:C_6$ x 2, $F_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 4, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 28: $D_{14}$

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

Order 20: $D_{10}$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 15: $C_{15}$ x 3

Order 14: $D_7$ x 2, $C_{14}$

Order 12: $C_2\times C_6$, $D_6$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1260: $D_{105}:C_6$ ($C_1$)

Order 630: $C_{105}:C_6$ ($C_2$), $C_{21}:C_{30}$ ($C_2$), $C_{105}:C_6$ ($C_2$)

Order 420: $D_5\times D_{21}$ ($C_3$)

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

Order 210: $C_5\times D_{21}$ ($C_6$), $D_5\times C_{21}$ ($C_6$), $C_{35}:C_6$ ($S_3$), $D_{105}$ ($C_6$)

Order 126: $C_{21}:C_6$ ($D_5$)

Order 105: $C_{105}$ ($C_2\times C_6$), $C_7:C_{15}$ ($D_6$)

Order 70: $C_7\times D_5$ ($C_3\times S_3$)

Order 63: $C_{21}:C_3$ ($D_{10}$)

Order 42: $D_{21}$ ($C_3\times D_5$)

Order 35: $C_{35}$ ($C_6\times S_3$)

Order 30: $C_3\times D_5$ ($F_7$)

Order 21: $C_{21}$ ($C_3\times D_{10}$), $C_7:C_3$ ($S_3\times D_5$)

Order 15: $C_{15}$ ($C_2\times F_7$)

Order 10: $D_5$ ($C_{21}:C_6$)

Order 7: $C_7$ ($C_{15}:D_6$)

Order 5: $C_5$ ($C_{42}:C_6$)

Order 3: $C_3$ ($D_5\times F_7$)

Order 1: $C_1$ ($D_{105}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1260: $D_{105}:C_6$ ($C_1$)

Order 630: $C_{105}:C_6$ ($C_2$), $C_{21}:C_{30}$ ($C_2$), $C_{105}:C_6$ ($C_2$)

Order 420: $D_5\times D_{21}$ ($C_3$)

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

Order 210: $C_5\times D_{21}$ ($C_6$), $D_5\times C_{21}$ ($C_6$), $C_{35}:C_6$ ($S_3$), $D_{105}$ ($C_6$)

Order 126: $C_{21}:C_6$ ($D_5$)

Order 105: $C_{105}$ ($C_2\times C_6$), $C_7:C_{15}$ ($D_6$)

Order 70: $C_7\times D_5$ ($C_3\times S_3$)

Order 63: $C_{21}:C_3$ ($D_{10}$)

Order 42: $D_{21}$ ($C_3\times D_5$)

Order 35: $C_{35}$ ($C_6\times S_3$)

Order 30: $C_3\times D_5$ ($F_7$)

Order 21: $C_{21}$ ($C_3\times D_{10}$), $C_7:C_3$ ($S_3\times D_5$)

Order 15: $C_{15}$ ($C_2\times F_7$)

Order 10: $D_5$ ($C_{21}:C_6$)

Order 7: $C_7$ ($C_{15}:D_6$)

Order 5: $C_5$ ($C_{42}:C_6$)

Order 3: $C_3$ ($D_5\times F_7$)

Order 1: $C_1$ ($D_{105}:C_6$)

Series

Derived series

$D_{105}:C_6$

$\rhd$

$C_{105}$

$\rhd$

$C_1$

Chief series

$D_{105}:C_6$

$\rhd$

$C_{105}:C_6$

$\rhd$

$C_{105}:C_3$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{105}:C_6$

$\rhd$

$C_{105}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 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 $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.