Abstract group 256.7667: $C_2^6:C_4$

• Label: 256.7667
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{18} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $32$
• Rank: $4$

Group information

Description:	$C_2^6:C_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^{16}.C_2^3.\PSL(2,7)$, of order \(88080384\)\(\medspace = 2^{22} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4
Elements	1	127	128	256
Conjugacy classes	1	71	16	88
Divisions	1	71	8	80
Autjugacy classes	1	3	1	5

Dimension	1	2
Irr. complex chars.	32	56	88
Irr. rational chars.	16	64	80

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$32$
Rank:	$4$
Inequivalent generating quadruples:	$15$

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}=g^{4}= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & 0 & 0 & \alpha \\ 0 & 1 & 0 & 0 \\ 0 & \alpha & 1 & 0 \\ \alpha & 0 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}\alpha & \alpha & 0 & \alpha^{2} \\ \alpha^{2} & 1 & 0 & \alpha^{2} \\ \alpha & \alpha^{2} & 1 & 0 \\ \alpha^{2} & \alpha & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \alpha & \alpha^{2} & 1 & \alpha \\ 1 & 0 & 0 & 0 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha^{2} & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & \alpha & 1 & 1 \\ 1 & \alpha^{2} & 0 & 0 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & \alpha^{2} \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ \alpha^{2} & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 0 & 5 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 12 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Transitive group:	32T5587				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $C_4$	$(C_2^5:C_4)$ $\,\rtimes\,$ $C_2$	$(C_2^4:C_4)$ $\,\rtimes\,$ $C_2^2$	$(C_2^3\times C_4)$ $\,\rtimes\,$ $C_2^3$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $D_4$	$C_2^7$ . $C_2$	$C_2^6$ . $C_2^2$	$C_2^5$ . $C_2^3$	all 16
Aut. group:	$\Aut(C_4^2.C_2^2)$

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{24}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{12}$
Commutator length:	$1$

Subgroups

There are 30559 subgroups in 15431 conjugacy classes, 687 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:C_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 256: $C_2^6:C_4$

Order 128: $C_2^5:C_4$ x 14, $C_2^7$

Order 64: $C_2^6$ x 71, $C_2^4:C_4$ x 56, $C_2^4:C_4$ x 28

Order 32: $C_2^5$ x 1379, $C_2^3:C_4$ x 168, $C_2^3\times C_4$ x 8

Order 16: $C_2^4$ x 6011, $C_2^2:C_4$ x 112, $C_2^2\times C_4$ x 56

Order 8: $C_2^3$ x 6011, $C_2\times C_4$ x 56

Order 4: $C_2^2$ x 1379, $C_4$ x 8

Order 2: $C_2$ x 71

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^6:C_4$

Order 128: $C_2^7$, $C_2^5:C_4$

Order 64: $C_2^6$ x 3, $C_2^4:C_4$, $C_2^4:C_4$

Order 32: $C_2^5$ x 6, $C_2^3\times C_4$, $C_2^3:C_4$

Order 16: $C_2^4$ x 10, $C_2^2:C_4$, $C_2^2\times C_4$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^6:C_4$ ($C_1$)

Order 128: $C_2^5:C_4$ ($C_2$) x 14, $C_2^7$ ($C_2$)

Order 64: $C_2^4:C_4$ ($C_2^2$) x 28, $C_2^6$ ($C_4$) x 8, $C_2^6$ ($C_2^2$) x 7

Order 32: $C_2^5$ ($D_4$) x 56, $C_2^5$ ($C_2\times C_4$) x 28, $C_2^3\times C_4$ ($C_2^3$) x 8, $C_2^5$ ($C_2^3$) x 7

Order 16: $C_2^4$ ($C_2^2:C_4$) x 112, $C_2^4$ ($C_2\times D_4$) x 84, $C_2^4$ ($C_2^2\times C_4$) x 14, $C_2^4$ ($C_2^4$)

Order 8: $C_2^3$ ($C_2^2\wr C_2$) x 112, $C_2^3$ ($C_2^3:C_4$) x 84, $C_2^3$ ($C_2^2\times D_4$) x 14, $C_2^3$ ($C_2^3\times C_4$)

Order 4: $C_2^2$ ($C_2^4:C_4$) x 56, $C_2^2$ ($C_2^3:D_4$) x 28, $C_2^2$ ($C_2^4:C_4$) x 7

Order 2: $C_2$ ($C_2^3\wr C_2$) x 8, $C_2$ ($C_2^5:C_4$) x 7

Order 1: $C_1$ ($C_2^6:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^6:C_4$ ($C_1$)

Order 128: $C_2^7$ ($C_2$), $C_2^5:C_4$ ($C_2$)

Order 64: $C_2^6$ ($C_2^2$), $C_2^6$ ($C_4$), $C_2^4:C_4$ ($C_2^2$)

Order 32: $C_2^5$ ($C_2^3$), $C_2^5$ ($D_4$), $C_2^5$ ($C_2\times C_4$), $C_2^3\times C_4$ ($C_2^3$)

Order 16: $C_2^4$ ($C_2^2:C_4$), $C_2^4$ ($C_2^4$), $C_2^4$ ($C_2\times D_4$), $C_2^4$ ($C_2^2\times C_4$)

Order 8: $C_2^3$ ($C_2^2\times D_4$), $C_2^3$ ($C_2^3\times C_4$), $C_2^3$ ($C_2^2\wr C_2$), $C_2^3$ ($C_2^3:C_4$)

Order 4: $C_2^2$ ($C_2^4:C_4$), $C_2^2$ ($C_2^3:D_4$), $C_2^2$ ($C_2^4:C_4$)

Order 2: $C_2$ ($C_2^3\wr C_2$), $C_2$ ($C_2^5:C_4$)

Order 1: $C_1$ ($C_2^6:C_4$)

Series

Derived series

$C_2^6:C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^6:C_4$

$\rhd$

$C_2^5:C_4$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6:C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$C_2^6:C_4$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.