Abstract group 768.1084399: $C_2^7:C_6$

• Label: 768.1084399
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $14$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^7:C_6$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:C_3.C_2^4.C_2^4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	143	32	112	352	128	768
Conjugacy classes	1	29	2	6	14	4	56
Divisions	1	29	1	6	7	2	46
Autjugacy classes	1	12	1	3	4	1	22

Dimension	1	2	3	4	6
Irr. complex chars.	24	6	8	0	18	56
Irr. rational chars.	8	10	8	2	18	46

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$10920$

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:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{4}=c^{2}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,6,8)(3,4,5), (1,2)(3,8)(4,6)(5,7)(9,10)(11,12), (1,3)(2,6)(4,5)(7,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 1 & 0 & 1 \\ \alpha^{2} & \alpha^{2} & 1 & \alpha \\ 1 & \alpha & \alpha^{2} & 1 \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 1 & \alpha & 1 \\ \alpha & \alpha & 1 & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 1 & 1 & \alpha \\ 1 & \alpha^{2} & 0 & \alpha \\ \alpha & \alpha^{2} & \alpha^{2} & \alpha \\ \alpha^{2} & 1 & \alpha^{2} & \alpha \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & 0 & \alpha^{2} \\ \alpha & \alpha & 1 & \alpha^{2} \\ \alpha^{2} & 1 & \alpha & \alpha^{2} \\ 1 & \alpha & 1 & 0 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ \alpha^{2} & 0 & \alpha & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 1 & 0 & 1 \\ \alpha & \alpha & 0 & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & \alpha & 0 & 1 \\ \alpha^{2} & \alpha & \alpha & \alpha \\ 1 & \alpha & \alpha^{2} & \alpha \\ \alpha^{2} & 1 & 1 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & \alpha^{2} & 0 & \alpha^{2} \\ 1 & \alpha^{2} & 0 & \alpha \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Transitive group:	24T2447	24T2448	24T2451	24T2453	all 11
Direct product:	$C_2$ $\, \times\, $ $(C_2\wr C_6)$
Semidirect product:	$C_2^7$ $\,\rtimes\,$ $C_6$	$(C_2^3:A_4)$ $\,\rtimes\,$ $D_4$	$(D_4\times C_2^3)$ $\,\rtimes\,$ $A_4$	$(C_2^5:A_4)$ $\,\rtimes\,$ $C_2$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $(D_4\times A_4)$	$C_2^5$ . $(C_2\times A_4)$	$C_2$ . $(C_2^5:A_4)$	$C_2^6$ . $(C_2\times C_6)$	all 11

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 33614 subgroups in 5670 conjugacy classes, 65 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4:A_4$
Commutator:	$G' \simeq$ $C_2^5$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^3\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5:D_4$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^7:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^2\times A_4$
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: $C_2^7:C_6$

Order 384: $C_2\wr C_6$ x 4, $C_2^5:C_{12}$, $C_2^5:A_4$, $C_2^5:A_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^4:A_4$ x 4, $C_2^4:C_{12}$ x 2, $C_2^5:C_6$

Order 128: $C_2^3\wr C_2$ x 8, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $C_2^7$

Order 96: $C_2^3:A_4$ x 5, $C_2^3\times A_4$ x 4, $D_4\times A_4$ x 4, $C_2^3:A_4$ x 2, $C_2^3:C_{12}$

Order 64: $C_2^3:D_4$ x 40, $C_2^6$ x 27, $C_2^4:C_4$ x 20, $C_2^4:C_4$ x 7, $D_4\times C_2^3$ x 3

Order 48: $C_2^2\times A_4$ x 23, $C_4\times A_4$ x 2, $C_2^2:A_4$, $C_6\times D_4$

Order 32: $C_2^5$ x 466, $C_2^2\wr C_2$ x 76, $C_2^3:C_4$ x 56, $C_2^2\times D_4$ x 40, $C_2^3\times C_4$ x 3

Order 24: $C_2\times A_4$ x 21, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$

Order 16: $C_2^4$ x 2025, $C_2\times D_4$ x 94, $C_2^2:C_4$ x 56, $C_2^2\times C_4$ x 20

Order 12: $C_2\times C_6$ x 9, $A_4$ x 3, $C_{12}$ x 2

Order 8: $C_2^3$ x 2033, $D_4$ x 40, $C_2\times C_4$ x 24

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 475, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 29

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2^7:C_6$

Order 384: $C_2\wr C_6$, $C_2^5:C_{12}$, $C_2^5:A_4$, $C_2^5:A_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^2\wr C_3$ x 3, $C_2^4:A_4$ x 2, $C_2^4:C_{12}$, $C_2^5:C_6$

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 3, $C_2^3:A_4$, $D_4\times A_4$, $C_2^3:C_{12}$

Order 64: $C_2^3:D_4$ x 12, $C_2^6$ 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 8, $C_2^2:A_4$, $C_6\times D_4$, $C_4\times A_4$

Order 32: $C_2^5$ x 75, $C_2^3:C_4$ x 20, $C_2^2\times D_4$ x 15, $C_2^2\wr C_2$ x 10, $C_2^3\times C_4$ x 3

Order 24: $C_2\times A_4$ x 7, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_3\times D_4$

Order 16: $C_2^4$ x 207, $C_2\times D_4$ x 21, $C_2^2\times C_4$ x 12, $C_2^2:C_4$ x 11

Order 12: $C_2\times C_6$ x 5, $A_4$ x 2, $C_{12}$

Order 8: $C_2^3$ x 211, $C_2\times C_4$ x 12, $D_4$ x 6

Order 6: $C_6$ x 4

Order 4: $C_2^2$ x 79, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 12

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2^7:C_6$ ($C_1$)

Order 384: $C_2\wr C_6$ ($C_2$) x 4, $C_2^5:C_{12}$ ($C_2$), $C_2^5:A_4$ ($C_2$), $C_2^5:A_4$ ($C_2$)

Order 256: $C_2^5:D_4$ ($C_3$)

Order 192: $C_2^2\wr C_3$ ($C_2^2$) x 3, $C_2^4:C_{12}$ ($C_2^2$) x 2, $C_2^4:A_4$ ($C_2^2$) x 2

Order 128: $C_2^3\wr C_2$ ($C_6$) x 4, $C_2^7$ ($C_6$), $C_2^4:D_4$ ($C_6$), $C_2^5:C_4$ ($C_6$)

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

Order 64: $C_2^6$ ($C_2\times C_6$) x 3, $C_2^4:C_4$ ($C_2\times C_6$) x 2, $C_2^3:D_4$ ($C_2\times C_6$) x 2, $D_4\times C_2^3$ ($A_4$)

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

Order 32: $C_2^2\times D_4$ ($C_2\times A_4$) x 4, $C_2^5$ ($C_2\times A_4$) x 2, $C_2^5$ ($C_3\times D_4$) x 2, $C_2^5$ ($C_2^2\times C_6$), $C_2^3\times C_4$ ($C_2\times A_4$)

Order 16: $C_2^4$ ($C_2^2\times A_4$) x 5, $C_2^2\times C_4$ ($C_2^2\times A_4$) x 2, $C_2^4$ ($C_6\times D_4$)

Order 8: $C_2^3$ ($D_4\times A_4$) x 2, $C_2^3$ ($C_2^3:A_4$), $C_2^3$ ($C_2^3\times A_4$)

Order 4: $C_2^2$ ($C_2^4:A_4$) x 3, $C_2^2$ ($C_2^5:C_6$)

Order 2: $C_2$ ($C_2\wr C_6$) x 2, $C_2$ ($C_2^5:A_4$)

Order 1: $C_1$ ($C_2^7:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2^7:C_6$ ($C_1$)

Order 384: $C_2\wr C_6$ ($C_2$), $C_2^5:C_{12}$ ($C_2$), $C_2^5:A_4$ ($C_2$), $C_2^5:A_4$ ($C_2$)

Order 256: $C_2^5:D_4$ ($C_3$)

Order 192: $C_2^2\wr C_3$ ($C_2^2$) x 2, $C_2^4:C_{12}$ ($C_2^2$), $C_2^4:A_4$ ($C_2^2$)

Order 128: $C_2^7$ ($C_6$), $C_2^4:D_4$ ($C_6$), $C_2^3\wr C_2$ ($C_6$), $C_2^5:C_4$ ($C_6$)

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

Order 64: $C_2^6$ ($C_2\times C_6$) x 2, $C_2^4:C_4$ ($C_2\times C_6$), $D_4\times C_2^3$ ($A_4$), $C_2^3:D_4$ ($C_2\times C_6$)

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

Order 32: $C_2^5$ ($C_2\times A_4$) x 2, $C_2^5$ ($C_2^2\times C_6$), $C_2^5$ ($C_3\times D_4$), $C_2^2\times D_4$ ($C_2\times A_4$), $C_2^3\times C_4$ ($C_2\times A_4$)

Order 16: $C_2^4$ ($C_2^2\times A_4$) x 3, $C_2^4$ ($C_6\times D_4$), $C_2^2\times C_4$ ($C_2^2\times A_4$)

Order 8: $C_2^3$ ($C_2^3:A_4$), $C_2^3$ ($C_2^3\times A_4$), $C_2^3$ ($D_4\times A_4$)

Order 4: $C_2^2$ ($C_2^4:A_4$) x 2, $C_2^2$ ($C_2^5:C_6$)

Order 2: $C_2$ ($C_2\wr C_6$), $C_2$ ($C_2^5:A_4$)

Order 1: $C_1$ ($C_2^7:C_6$)

Series

Derived series

$C_2^7:C_6$

$\rhd$

$C_2^5$

$\rhd$

$C_1$

Chief series

$C_2^7:C_6$

$\rhd$

$C_2\wr C_6$

$\rhd$

$C_2^2\wr C_3$

$\rhd$

$C_2^3:A_4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^7:C_6$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 47 larger groups in the database.

This group is a maximal quotient of 13 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 $46 \times 46$ rational character table.