Abstract group 768.1090187: $C_2^3:\GL(2,\mathbb{Z}/4)$

• Label: 768.1090187
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^3:\GL(2,\mathbb{Z}/4)$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4^2.C_2^5.C_2^2$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6
Elements	1	175	32	336	224	768
Conjugacy classes	1	33	1	14	7	56
Divisions	1	33	1	14	5	54
Autjugacy classes	1	12	1	3	3	20

Dimension	1	2	3	4	6
Irr. complex chars.	8	10	24	0	14	56
Irr. rational chars.	8	6	24	2	14	54

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$5880$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{4}=c^{6}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha & \alpha & 1 & 0 \\ \alpha^{2} & 1 & \alpha^{2} & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 1 & \alpha & \alpha \\ \alpha^{2} & \alpha^{2} & \alpha & 0 \\ 1 & 1 & 0 & \alpha^{2} \\ 0 & 0 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha & \alpha & 1 & \alpha \\ \alpha^{2} & 1 & \alpha^{2} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & \alpha^{2} & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 1 & \alpha^{2} & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & \alpha^{2} \\ 0 & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha & \alpha & 1 & 1 \\ \alpha^{2} & 1 & \alpha^{2} & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha^{2} & 0 & 0 \\ 1 & 1 & \alpha & 0 \\ \alpha^{2} & \alpha & 1 & \alpha^{2} \\ \alpha^{2} & 1 & \alpha & 0 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ 0 & 1 & 0 & 0 \\ \alpha & \alpha^{2} & 0 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Transitive group:	24T1932	24T1974	24T2204	24T2469	all 11
Direct product:	$C_2$ $\, \times\, $ $(C_2\wr S_3)$
Semidirect product:	$C_2^7$ $\,\rtimes\,$ $S_3$	$C_2^5$ $\,\rtimes\,$ $S_4$	$C_2^6$ $\,\rtimes\,$ $D_6$	$(C_2^4.S_4)$ $\,\rtimes\,$ $C_2$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $D_6$	$C_2^4$ . $(C_2\times S_4)$	$C_2^5$ . $(C_2\times D_6)$	$C_2$ . $(C_2^2\wr S_3)$	all 8
Aut. group:	$\Aut(C_6.C_4^2)$	$\Aut(C_2^4:C_6)$

Elements of the group are displayed as matrices in $\GL_{4}(\F_{4})$.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$2$

Subgroups

There are 40588 subgroups in 6896 conjugacy classes, 59 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^3:S_4$
Commutator:	$G' \simeq$ $C_2^3:A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\wr S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^7$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^3:\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^6$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$

Order 384: $C_2\wr S_3$ x 4, $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2^4.S_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^3.S_4$ x 2

Order 128: $C_2^3\wr C_2$ x 16, $C_2^4:D_4$ x 7, $C_2^5:C_4$ x 7, $C_2^7$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 12, $C_2^3:A_4$ x 5, $C_2^3\times A_4$ x 4, $C_2^2\times S_4$ x 3, $C_2^2.S_4$ x 3, $C_2^2:S_4$ x 2

Order 64: $C_2^3:D_4$ x 112, $C_2^4:C_4$ x 56, $C_2^6$ x 31, $C_2^4:C_4$ x 21, $D_4\times C_2^3$ x 7

Order 48: $C_2^2\times A_4$ x 22, $C_2\times S_4$ x 12, $A_4:C_4$ x 6, $C_2^2:A_4$, $C_6:D_4$

Order 32: $C_2^5$ x 495, $C_2^2\wr C_2$ x 224, $C_2^3:C_4$ x 168, $C_2^2\times D_4$ x 112, $C_2^3\times C_4$ x 7

Order 24: $C_2\times A_4$ x 22, $S_4$ x 6, $C_3:D_4$ x 4, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$

Order 16: $C_2^4$ x 2102, $C_2\times D_4$ x 280, $C_2^2:C_4$ x 168, $C_2^2\times C_4$ x 56

Order 12: $C_2\times C_6$ x 5, $D_6$ x 4, $A_4$ x 4, $C_3:C_4$ x 2

Order 8: $C_2^3$ x 2128, $D_4$ x 112, $C_2\times C_4$ x 70

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

Order 4: $C_2^2$ x 516, $C_4$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 33

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$

Order 384: $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2\wr S_3$, $C_2^4.S_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^2\wr C_3$ x 3, $C_2^3:S_4$ x 2, $C_2^3.S_4$, $C_2\times \GL(2,\mathbb{Z}/4)$

Order 128: $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $C_2^3\wr C_2$ x 2, $C_2^7$

Order 96: $C_2^3:A_4$ x 3, $C_2^3\times A_4$ x 2, $C_2^2:S_4$, $C_2^2\times S_4$, $\GL(2,\mathbb{Z}/4)$, $C_2^2.S_4$

Order 64: $C_2^3:D_4$ x 16, $C_2^6$ x 11, $C_2^4:C_4$ x 6, $C_2^4:C_4$ x 6, $D_4\times C_2^3$ x 3

Order 48: $C_2^2\times A_4$ x 6, $C_2\times S_4$ x 2, $C_2^2:A_4$, $C_6:D_4$, $A_4:C_4$

Order 32: $C_2^5$ x 69, $C_2^3:C_4$ x 24, $C_2^2\times D_4$ x 19, $C_2^2\wr C_2$ x 15, $C_2^3\times C_4$ x 3

Order 24: $C_2\times A_4$ x 6, $C_3:D_4$, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$, $S_4$

Order 16: $C_2^4$ x 179, $C_2\times D_4$ x 30, $C_2^2:C_4$ x 15, $C_2^2\times C_4$ x 12

Order 12: $C_2\times C_6$ x 3, $D_6$ x 2, $A_4$ x 2, $C_3:C_4$

Order 8: $C_2^3$ x 183, $C_2\times C_4$ x 12, $D_4$ x 10

Order 6: $C_6$ x 3, $S_3$

Order 4: $C_2^2$ x 73, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 12

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 384: $C_2\wr S_3$ ($C_2$) x 4, $C_2^5:A_4$ ($C_2$), $C_2^2\wr S_3$ ($C_2$), $C_2^4.S_4$ ($C_2$)

Order 192: $C_2^2\wr C_3$ ($C_2^2$) x 3, $C_2^3:S_4$ ($C_2^2$) x 2, $C_2^3.S_4$ ($C_2^2$) x 2

Order 128: $C_2^7$ ($S_3$)

Order 96: $C_2^3:A_4$ ($D_4$) x 2, $C_2^3:A_4$ ($C_2^3$)

Order 64: $C_2^6$ ($D_6$) x 3

Order 48: $C_2^2:A_4$ ($C_2\times D_4$)

Order 32: $C_2^5$ ($S_4$) x 3, $C_2^5$ ($C_3:D_4$) x 2, $C_2^5$ ($C_2\times D_6$)

Order 16: $C_2^4$ ($C_2\times S_4$) x 9, $C_2^4$ ($C_6:D_4$)

Order 8: $C_2^3$ ($\GL(2,\mathbb{Z}/4)$) x 6, $C_2^3$ ($C_2^2\times S_4$) x 3, $C_2^3$ ($C_2^2:S_4$)

Order 4: $C_2^2$ ($C_2^3:S_4$) x 3, $C_2^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 3

Order 2: $C_2$ ($C_2\wr S_3$) x 2, $C_2$ ($C_2^2\wr S_3$)

Order 1: $C_1$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 384: $C_2^5:A_4$ ($C_2$), $C_2^2\wr S_3$ ($C_2$), $C_2\wr S_3$ ($C_2$), $C_2^4.S_4$ ($C_2$)

Order 192: $C_2^2\wr C_3$ ($C_2^2$) x 2, $C_2^3:S_4$ ($C_2^2$), $C_2^3.S_4$ ($C_2^2$)

Order 128: $C_2^7$ ($S_3$)

Order 96: $C_2^3:A_4$ ($C_2^3$), $C_2^3:A_4$ ($D_4$)

Order 64: $C_2^6$ ($D_6$) x 2

Order 48: $C_2^2:A_4$ ($C_2\times D_4$)

Order 32: $C_2^5$ ($C_3:D_4$), $C_2^5$ ($C_2\times D_6$), $C_2^5$ ($S_4$)

Order 16: $C_2^4$ ($C_2\times S_4$) x 2, $C_2^4$ ($C_6:D_4$)

Order 8: $C_2^3$ ($C_2^2:S_4$), $C_2^3$ ($C_2^2\times S_4$), $C_2^3$ ($\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($C_2^3:S_4$) x 2, $C_2^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^2\wr S_3$), $C_2$ ($C_2\wr S_3$)

Order 1: $C_1$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$C_2^3:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^3:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^3:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^5:A_4$

$\rhd$

$C_2^2\wr C_3$

$\rhd$

$C_2^3:A_4$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^3:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^3:A_4$

$\rhd$

$C_2^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 51 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.