Abstract group 960.6256: $C_5\times Q_8.D_{12}$

• Label: 960.6256
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $25$
• Trans deg.: $320$
• Rank: $2$

Group information

Description:	$C_5\times Q_8.D_{12}$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$A_4.C_2^6.C_2^2$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	40	60
Elements	1	3	8	76	4	24	48	12	32	32	304	96	192	128	960
Conjugacy classes	1	3	1	7	4	3	4	12	4	4	28	12	16	16	115
Divisions	1	3	1	7	1	3	2	3	2	1	7	3	2	2	38
Autjugacy classes	1	3	1	5	1	3	1	3	1	1	5	3	1	1	30

Dimension	1	2	3	4	6	8	12	16	24	32
Irr. complex chars.	20	45	20	25	5	0	0	0	0	0	115
Irr. rational chars.	4	3	4	10	1	4	4	6	1	1	38

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$320$
Rank:	$2$
Inequivalent generating pairs:	$54$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	${\langle a, b, c, d \mid b^{12}=d^{20}=1, a^{2}=d^{10}, c^{2}=d^{10}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $25$ $\langle(1,2,5,8)(3,9,11,7)(4,12,14,6)(10,16,13,15)(17,18)(19,20), (17,19,20,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 67 & 66 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 45 & 55 \\ 66 & 67 \end{array}\right), \left(\begin{array}{rr} 27 & 31 \\ 12 & 61 \end{array}\right), \left(\begin{array}{rr} 15 & 63 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 23 & 33 \\ 55 & 12 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/77\Z)$
Direct product:	$C_5$ $\, \times\, $ $(Q_8.D_{12})$
Semidirect product:	$C_{20}$ $\,\rtimes\,$ $(C_2.S_4)$ (2)	$C_4$ $\,\rtimes\,$ $(C_{10}.S_4)$ (2)			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8\times C_{20})$ . $S_3$	$(C_2\times C_{20})$ . $S_4$	$C_{10}$ . $(C_4:S_4)$	$C_{10}$ . $(C_4.S_4)$	all 24

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 498 subgroups in 134 conjugacy classes, 34 normal (26 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $C_2\times S_4$
Commutator:	$G' \simeq$ $C_2\times \SL(2,3)$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_{10}\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{20}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_5\times Q_8.D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:Q_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 960: $C_5\times Q_8.D_{12}$

Order 480: $C_2\times C_{10}.S_4$ x 2, $C_{20}\times \SL(2,3)$

Order 320: $C_{20}:Q_{16}$

Order 240: $C_{10}.S_4$ x 2, $C_{12}:C_{20}$, $C_{10}\times \SL(2,3)$

Order 192: $Q_8.D_{12}$

Order 160: $Q_8:C_{20}$ x 2, $C_{10}\times Q_{16}$ x 2, $C_4:C_{40}$, $C_{20}:Q_8$, $Q_8\times C_{20}$

Order 120: $C_6:C_{20}$ x 2, $C_2\times C_{60}$, $C_5\times \SL(2,3)$

Order 96: $C_2^2.S_4$ x 2, $C_4\times \SL(2,3)$

Order 80: $C_5\times Q_{16}$ x 4, $Q_8\times C_{10}$ x 3, $C_4:C_{20}$ x 3, $C_2\times C_{40}$ x 2, $C_4\times C_{20}$

Order 64: $C_4:Q_{16}$

Order 60: $C_{60}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 48: $C_2.S_4$ x 2, $C_2\times \SL(2,3)$, $C_{12}:C_4$

Order 40: $C_5\times Q_8$ x 6, $C_2\times C_{20}$ x 5, $C_{40}$ x 2

Order 32: $C_2\times Q_{16}$ x 2, $Q_8:C_4$ x 2, $C_4:Q_8$, $C_4\times Q_8$, $C_4:C_8$

Order 30: $C_{30}$ x 3

Order 24: $C_6:C_4$ x 2, $C_2\times C_{12}$, $\SL(2,3)$

Order 20: $C_{20}$ x 7, $C_2\times C_{10}$

Order 16: $Q_{16}$ x 4, $C_4:C_4$ x 3, $C_2\times Q_8$ x 3, $C_2\times C_8$ x 2, $C_4^2$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 5, $C_8$ x 2

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_5\times Q_8.D_{12}$

Order 480: $C_{20}\times \SL(2,3)$, $C_2\times C_{10}.S_4$

Order 320: $C_{20}:Q_{16}$

Order 240: $C_{12}:C_{20}$, $C_{10}\times \SL(2,3)$, $C_{10}.S_4$

Order 192: $Q_8.D_{12}$

Order 160: $C_4:C_{40}$, $Q_8:C_{20}$, $C_{10}\times Q_{16}$, $C_{20}:Q_8$, $Q_8\times C_{20}$

Order 120: $C_2\times C_{60}$, $C_6:C_{20}$, $C_5\times \SL(2,3)$

Order 96: $C_4\times \SL(2,3)$, $C_2^2.S_4$

Order 80: $C_4:C_{20}$ x 3, $Q_8\times C_{10}$ x 2, $C_5\times Q_{16}$ x 2, $C_2\times C_{40}$, $C_4\times C_{20}$

Order 64: $C_4:Q_{16}$

Order 60: $C_{60}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_2\times \SL(2,3)$, $C_2.S_4$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 4, $C_5\times Q_8$ x 4, $C_{40}$

Order 32: $C_2\times Q_{16}$, $C_4:Q_8$, $C_4\times Q_8$, $C_4:C_8$, $Q_8:C_4$

Order 30: $C_{30}$ x 3

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$

Order 20: $C_{20}$ x 5, $C_2\times C_{10}$

Order 16: $C_4:C_4$ x 3, $Q_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2\times C_8$, $C_4^2$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 4, $C_8$

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_5\times Q_8.D_{12}$ ($C_1$)

Order 480: $C_2\times C_{10}.S_4$ ($C_2$) x 2, $C_{20}\times \SL(2,3)$ ($C_2$)

Order 240: $C_{10}\times \SL(2,3)$ ($C_2^2$)

Order 192: $Q_8.D_{12}$ ($C_5$)

Order 160: $Q_8\times C_{20}$ ($S_3$)

Order 120: $C_5\times \SL(2,3)$ ($D_4$)

Order 96: $C_2^2.S_4$ ($C_{10}$) x 2, $C_4\times \SL(2,3)$ ($C_{10}$)

Order 80: $Q_8\times C_{10}$ ($D_6$)

Order 48: $C_2\times \SL(2,3)$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($S_4$), $C_5\times Q_8$ ($D_{12}$)

Order 32: $C_4\times Q_8$ ($C_5\times S_3$)

Order 24: $\SL(2,3)$ ($C_5\times D_4$)

Order 20: $C_{20}$ ($C_2.S_4$) x 2, $C_2\times C_{10}$ ($C_2\times S_4$)

Order 16: $C_2\times Q_8$ ($S_3\times C_{10}$)

Order 10: $C_{10}$ ($C_4.S_4$), $C_{10}$ ($C_2^2.S_4$), $C_{10}$ ($C_4:S_4$)

Order 8: $Q_8$ ($C_5\times D_{12}$), $C_2\times C_4$ ($C_5\times S_4$)

Order 5: $C_5$ ($Q_8.D_{12}$)

Order 4: $C_4$ ($C_{10}.S_4$) x 2, $C_2^2$ ($C_{10}\times S_4$)

Order 2: $C_2$ ($C_{20}.S_4$), $C_2$ ($C_2\times C_{10}.S_4$), $C_2$ ($C_{20}:S_4$)

Order 1: $C_1$ ($C_5\times Q_8.D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_5\times Q_8.D_{12}$ ($C_1$)

Order 480: $C_{20}\times \SL(2,3)$ ($C_2$), $C_2\times C_{10}.S_4$ ($C_2$)

Order 240: $C_{10}\times \SL(2,3)$ ($C_2^2$)

Order 192: $Q_8.D_{12}$ ($C_5$)

Order 160: $Q_8\times C_{20}$ ($S_3$)

Order 120: $C_5\times \SL(2,3)$ ($D_4$)

Order 96: $C_4\times \SL(2,3)$ ($C_{10}$), $C_2^2.S_4$ ($C_{10}$)

Order 80: $Q_8\times C_{10}$ ($D_6$)

Order 48: $C_2\times \SL(2,3)$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($S_4$), $C_5\times Q_8$ ($D_{12}$)

Order 32: $C_4\times Q_8$ ($C_5\times S_3$)

Order 24: $\SL(2,3)$ ($C_5\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_2\times S_4$), $C_{20}$ ($C_2.S_4$)

Order 16: $C_2\times Q_8$ ($S_3\times C_{10}$)

Order 10: $C_{10}$ ($C_4.S_4$), $C_{10}$ ($C_2^2.S_4$), $C_{10}$ ($C_4:S_4$)

Order 8: $Q_8$ ($C_5\times D_{12}$), $C_2\times C_4$ ($C_5\times S_4$)

Order 5: $C_5$ ($Q_8.D_{12}$)

Order 4: $C_2^2$ ($C_{10}\times S_4$), $C_4$ ($C_{10}.S_4$)

Order 2: $C_2$ ($C_{20}.S_4$), $C_2$ ($C_2\times C_{10}.S_4$), $C_2$ ($C_{20}:S_4$)

Order 1: $C_1$ ($C_5\times Q_8.D_{12}$)

Series

Derived series

$C_5\times Q_8.D_{12}$

$\rhd$

$C_2\times \SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_5\times Q_8.D_{12}$

$\rhd$

$C_2\times C_{10}.S_4$

$\rhd$

$C_{10}\times \SL(2,3)$

$\rhd$

$C_5\times \SL(2,3)$

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5\times Q_8.D_{12}$

$\rhd$

$C_2\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$C_2\times C_{20}$

Character theory

Complex character table

See the $115 \times 115$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.