Abstract group 384.17286: $C_2^6:C_6$

• Label: 384.17286
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \cdot 7 \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$C_2^6:C_6$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_2^{12}.\PSL(2,7)$, of order \(1376256\)\(\medspace = 2^{16} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	71	2	56	142	112	384
Conjugacy classes	1	36	2	7	72	14	132
Divisions	1	36	1	7	36	7	88
Autjugacy classes	1	3	1	1	3	1	10

Dimension	1	2	4
Irr. complex chars.	48	84	0	132
Irr. rational chars.	16	44	28	88

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$4800$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{12}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,6)(4,7)(5,8)(10,11), (1,3)(2,6)(4,5)(7,8)(9,10)(11,12), (2,7)(6,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 1 & 0 & 1 \\ 0 & \alpha^{2} & 0 & 0 \\ \alpha & 0 & \alpha^{2} & 0 \\ 1 & 1 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 1 \\ 0 & \alpha^{2} & 0 & 0 \\ 0 & \alpha^{2} & \alpha^{2} & \alpha \\ 1 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & 0 & 0 & \alpha \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & 1 \\ \alpha & 0 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & \alpha^{2} \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ \alpha^{2} & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 9 \\ 21 & 16 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 8 & 3 \\ 21 & 8 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_2^3\wr C_2)$
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $C_6$	$(C_2^3:D_4)$ $\,\rtimes\,$ $C_6$	$(C_2^3\times C_6)$ $\,\rtimes\,$ $D_4$	$C_2^4$ $\,\rtimes\,$ $(C_3\times D_4)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $(C_6\times D_4)$	$C_6$ . $(C_2^3:D_4)$	$C_2^5$ . $(C_2\times C_6)$	$C_2$ . $(C_2^5:C_6)$	all 13

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 6664 subgroups in 3422 conjugacy classes, 360 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_6$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_6$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\wr C_2$
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

Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 384: $C_2^6:C_6$

Order 192: $C_2^4:C_{12}$ x 7, $C_2^5:C_6$ x 7, $C_2^5\times C_6$

Order 128: $C_2^3\wr C_2$

Order 96: $C_2^4\times C_6$ x 35, $C_2^4:C_6$ x 28, $C_2^3:C_{12}$ x 21, $C_{12}:C_2^3$ x 7

Order 64: $C_2^4:C_4$ x 7, $C_2^3:D_4$ x 7, $C_2^6$

Order 48: $C_2^3\times C_6$ x 344, $C_6\times D_4$ x 42, $C_2^2:C_{12}$ x 42, $C_2^2\times C_{12}$ x 7

Order 32: $C_2^5$ x 35, $C_2^2\wr C_2$ x 28, $C_2^3:C_4$ x 21, $C_2^2\times D_4$ x 7

Order 24: $C_2^2\times C_6$ x 726, $C_3\times D_4$ x 28, $C_2\times C_{12}$ x 21

Order 16: $C_2^4$ x 344, $C_2^2:C_4$ x 42, $C_2\times D_4$ x 42, $C_2^2\times C_4$ x 7

Order 12: $C_2\times C_6$ x 350, $C_{12}$ x 7

Order 8: $C_2^3$ x 726, $D_4$ x 28, $C_2\times C_4$ x 21

Order 6: $C_6$ x 36

Order 4: $C_2^2$ x 350, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 36

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2^6:C_6$

Order 192: $C_2^4:C_{12}$, $C_2^5\times C_6$, $C_2^5:C_6$

Order 128: $C_2^3\wr C_2$

Order 96: $C_2^4\times C_6$ x 2, $C_{12}:C_2^3$, $C_2^4:C_6$, $C_2^3:C_{12}$

Order 64: $C_2^4:C_4$, $C_2^6$, $C_2^3:D_4$

Order 48: $C_2^3\times C_6$ x 5, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_2^2:C_{12}$

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

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

Order 16: $C_2^4$ x 5, $C_2^2:C_4$, $C_2\times D_4$, $C_2^2\times C_4$

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

Order 8: $C_2^3$ x 7, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2^6:C_6$ ($C_1$)

Order 192: $C_2^4:C_{12}$ ($C_2$) x 7, $C_2^5:C_6$ ($C_2$) x 7, $C_2^5\times C_6$ ($C_2$)

Order 128: $C_2^3\wr C_2$ ($C_3$)

Order 96: $C_2^3:C_{12}$ ($C_2^2$) x 21, $C_2^4\times C_6$ ($C_2^2$) x 7, $C_{12}:C_2^3$ ($C_2^2$) x 7

Order 64: $C_2^4:C_4$ ($C_6$) x 7, $C_2^3:D_4$ ($C_6$) x 7, $C_2^6$ ($C_6$)

Order 48: $C_2^3\times C_6$ ($D_4$) x 28, $C_2^3\times C_6$ ($C_2^3$) x 8, $C_2^2\times C_{12}$ ($C_2^3$) x 7

Order 32: $C_2^3:C_4$ ($C_2\times C_6$) x 21, $C_2^5$ ($C_2\times C_6$) x 7, $C_2^2\times D_4$ ($C_2\times C_6$) x 7

Order 24: $C_2^2\times C_6$ ($C_2\times D_4$) x 42, $C_2^2\times C_6$ ($C_2^4$)

Order 16: $C_2^4$ ($C_3\times D_4$) x 28, $C_2^4$ ($C_2^2\times C_6$) x 8, $C_2^2\times C_4$ ($C_2^2\times C_6$) x 7

Order 12: $C_2\times C_6$ ($C_2^2\wr C_2$) x 28, $C_2\times C_6$ ($C_2^2\times D_4$) x 7

Order 8: $C_2^3$ ($C_6\times D_4$) x 42, $C_2^3$ ($C_2^3\times C_6$)

Order 6: $C_6$ ($C_2^3:D_4$) x 7

Order 4: $C_2^2$ ($C_2^4:C_6$) x 28, $C_2^2$ ($C_{12}:C_2^3$) x 7

Order 3: $C_3$ ($C_2^3\wr C_2$)

Order 2: $C_2$ ($C_2^5:C_6$) x 7

Order 1: $C_1$ ($C_2^6:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2^6:C_6$ ($C_1$)

Order 192: $C_2^4:C_{12}$ ($C_2$), $C_2^5\times C_6$ ($C_2$), $C_2^5:C_6$ ($C_2$)

Order 128: $C_2^3\wr C_2$ ($C_3$)

Order 96: $C_2^4\times C_6$ ($C_2^2$), $C_{12}:C_2^3$ ($C_2^2$), $C_2^3:C_{12}$ ($C_2^2$)

Order 64: $C_2^4:C_4$ ($C_6$), $C_2^6$ ($C_6$), $C_2^3:D_4$ ($C_6$)

Order 48: $C_2^3\times C_6$ ($C_2^3$) x 2, $C_2^3\times C_6$ ($D_4$), $C_2^2\times C_{12}$ ($C_2^3$)

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

Order 24: $C_2^2\times C_6$ ($C_2^4$), $C_2^2\times C_6$ ($C_2\times D_4$)

Order 16: $C_2^4$ ($C_2^2\times C_6$) x 2, $C_2^4$ ($C_3\times D_4$), $C_2^2\times C_4$ ($C_2^2\times C_6$)

Order 12: $C_2\times C_6$ ($C_2^2\times D_4$), $C_2\times C_6$ ($C_2^2\wr C_2$)

Order 8: $C_2^3$ ($C_2^3\times C_6$), $C_2^3$ ($C_6\times D_4$)

Order 6: $C_6$ ($C_2^3:D_4$)

Order 4: $C_2^2$ ($C_{12}:C_2^3$), $C_2^2$ ($C_2^4:C_6$)

Order 3: $C_3$ ($C_2^3\wr C_2$)

Order 2: $C_2$ ($C_2^5:C_6$)

Order 1: $C_1$ ($C_2^6:C_6$)

Series

Derived series

$C_2^6:C_6$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^6:C_6$

$\rhd$

$C_2^5\times C_6$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^6:C_6$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_2^6:C_6$

Supergroups

This group is a maximal subgroup of 12 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 $132 \times 132$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.