Abstract group 768.375391: $C_2^6:D_6$

• Label: 768.375391
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \cdot 3 \cdot 7 \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $5$

Group information

Description:	$C_2^6:D_6$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^{15}.\PSL(2,7)\times S_3$, of order \(33030144\)\(\medspace = 2^{19} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	287	2	224	142	112	768
Conjugacy classes	1	73	1	14	36	7	132
Divisions	1	73	1	14	36	7	132
Autjugacy classes	1	6	1	2	3	1	14

Dimension	1	2	4
Irr. complex chars.	32	72	28	132
Irr. rational chars.	32	72	28	132

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$5$
Inequivalent generating 5-tuples:	$2380800$

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{4}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,3)(8,9)(10,12)(11,14)(13,15), (1,2)(3,4), (1,2)(3,4)(9,12)(13,14), (1,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 9 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 20 \\ 6 & 5 \end{array}\right), \left(\begin{array}{rr} 5 & 12 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 12 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$S_3$ $\, \times\, $ $(C_2^3\wr C_2)$
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $D_6$	$C_2^4$ $\,\rtimes\,$ $(S_3\times D_4)$	$(C_2^6:S_3)$ $\,\rtimes\,$ $C_2$	$(C_2^3\times D_6)$ $\,\rtimes\,$ $D_4$	all 46
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $(C_2^3:D_4)$	$C_2^3$ . $(D_4\times D_6)$	$C_2^5$ . $(C_2\times D_6)$	$C_6$ . $(C_2^4:D_4)$	all 18
Aut. group:	$\Aut(C_6.C_4^2)$	$\Aut(C_4^2:S_3)$	$\Aut(C_2^3.D_6)$	$\Aut(D_6.D_4)$	all 9

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{13}$
Commutator length:	$1$

Subgroups

There are 123828 subgroups in 32890 conjugacy classes, 1123 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^3\times D_6$
Commutator:	$G' \simeq$ $C_2^2\times C_6$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_6$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^6:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^5$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 768: $C_2^6:D_6$

Order 384: $C_2^4:D_{12}$ x 7, $C_2^5.D_6$ x 7, $C_2^5:D_6$ x 7, $C_2^5.D_6$ x 7, $D_6\times C_2^5$, $C_2^6:C_6$, $C_2^6:S_3$

Order 256: $C_2^5:D_4$

Order 192: $D_6\times C_2^4$ x 70, $C_2^4:D_6$ x 56, $C_2^3.D_{12}$ x 42, $C_2^4:D_6$ x 21, $C_2^3:D_{12}$ x 21, $C_2^4.D_6$ x 21, $C_2^4:C_{12}$ x 7, $C_2^4.D_6$ x 7, $C_{12}:C_2^4$ x 7, $C_2^5:C_6$ x 7, $C_2^4:D_6$ x 7, $C_2^5\times C_6$

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: $C_2^3\times D_6$ x 1345, $C_2^2:D_{12}$ x 84, $D_6.D_4$ x 84, $D_4\times D_6$ x 84, $D_6:D_4$ x 84, $C_2^2.D_{12}$ x 42, $C_2^4\times C_6$ x 35, $C_2^4:C_6$ x 28, $C_2^4:S_3$ x 28, $C_2^3:C_{12}$ x 21, $C_2^3.D_6$ x 21, $C_2^3:D_6$ x 14, $C_{12}:C_2^3$ x 7, $C_2^2\times D_{12}$ x 7, $C_{12}:C_2^3$ x 7

Order 64: $C_2^3:D_4$ x 112, $C_2^6$ x 71, $C_2^4:C_4$ x 56, $C_2^4:C_4$ x 21, $D_4\times C_2^3$ x 7

Order 48: $C_2^2\times D_6$ x 5683, $C_2^3\times C_6$ x 344, $S_3\times D_4$ x 112, $C_6:D_4$ x 84, $D_6:C_4$ x 84, $C_6\times D_4$ x 42, $C_2\times D_{12}$ x 42, $C_4\times D_6$ x 42, $C_2^2:C_{12}$ x 42, $C_6.D_4$ x 42, $C_2^2\times C_{12}$ x 7, $C_6.C_2^3$ x 7

Order 32: $C_2^5$ x 1380, $C_2^2\wr C_2$ 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: $C_2\times D_6$ x 5327, $C_2^2\times C_6$ x 726, $C_3:D_4$ x 56, $D_{12}$ x 28, $C_4\times S_3$ x 28, $C_3\times D_4$ x 28, $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21

Order 16: $C_2^4$ x 6027, $C_2\times D_4$ x 280, $C_2^2:C_4$ x 168, $C_2^2\times C_4$ x 56

Order 12: $D_6$ x 1051, $C_2\times C_6$ x 350, $C_{12}$ x 7, $C_3:C_4$ x 7

Order 8: $C_2^3$ x 6053, $D_4$ x 112, $C_2\times C_4$ x 70

Order 6: $S_3$ x 37, $C_6$ x 36

Order 4: $C_2^2$ x 1401, $C_4$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 73

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2^6:D_6$

Order 384: $C_2^4:D_{12}$, $C_2^5.D_6$, $D_6\times C_2^5$, $C_2^5:D_6$, $C_2^6:C_6$, $C_2^6:S_3$, $C_2^5.D_6$

Order 256: $C_2^5:D_4$

Order 192: $D_6\times C_2^4$ x 3, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^5\times C_6$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

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

Order 96: $C_2^3\times D_6$ x 8, $C_2^4\times C_6$ x 2, $C_2^3:D_6$ x 2, $C_2^2:D_{12}$, $D_6.D_4$, $C_{12}:C_2^3$, $D_4\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^4:C_6$, $C_2^3:C_{12}$, $C_2^4:S_3$, $C_2^3.D_6$, $D_6:D_4$, $C_2^2.D_{12}$

Order 64: $C_2^3:D_4$ x 5, $C_2^6$ x 4, $C_2^4:C_4$ x 3, $D_4\times C_2^3$, $C_2^4:C_4$

Order 48: $C_2^2\times D_6$ x 14, $C_2^3\times C_6$ x 5, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_2^2:C_{12}$, $C_6.D_4$, $D_6:C_4$

Order 32: $C_2^5$ x 10, $C_2^2\times D_4$ x 5, $C_2^2\wr C_2$ x 4, $C_2^3:C_4$ x 4, $C_2^3\times C_4$

Order 24: $C_2\times D_6$ x 12, $C_2^2\times C_6$ x 7, $C_3:D_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 16: $C_2^4$ x 19, $C_2\times D_4$ x 5, $C_2^2:C_4$ x 3, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 7, $C_2\times C_6$ x 5, $C_{12}$, $C_3:C_4$

Order 8: $C_2^3$ x 19, $D_4$ x 4, $C_2\times C_4$ x 3

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

Order 4: $C_2^2$ x 12, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2^6:D_6$ ($C_1$)

Order 384: $C_2^4:D_{12}$ ($C_2$) x 7, $C_2^5.D_6$ ($C_2$) x 7, $C_2^5:D_6$ ($C_2$) x 7, $C_2^5.D_6$ ($C_2$) x 7, $D_6\times C_2^5$ ($C_2$), $C_2^6:C_6$ ($C_2$), $C_2^6:S_3$ ($C_2$)

Order 192: $C_2^3.D_{12}$ ($C_2^2$) x 42, $C_2^4:D_6$ ($C_2^2$) x 21, $C_2^3:D_{12}$ ($C_2^2$) x 21, $C_2^4.D_6$ ($C_2^2$) x 21, $D_6\times C_2^4$ ($C_2^2$) x 14, $C_2^4:C_{12}$ ($C_2^2$) x 7, $C_2^4.D_6$ ($C_2^2$) x 7, $C_{12}:C_2^4$ ($C_2^2$) x 7, $C_2^5:C_6$ ($C_2^2$) x 7, $C_2^4:D_6$ ($C_2^2$) x 7, $C_2^5\times C_6$ ($C_2^2$)

Order 128: $C_2^3\wr C_2$ ($S_3$)

Order 96: $C_2^3\times D_6$ ($D_4$) x 56, $C_2^2.D_{12}$ ($C_2^3$) x 42, $C_2^3\times D_6$ ($C_2^3$) x 29, $C_2^3:C_{12}$ ($C_2^3$) x 21, $C_2^3.D_6$ ($C_2^3$) x 21, $C_2^3:D_6$ ($C_2^3$) x 14, $C_2^4\times C_6$ ($C_2^3$) x 7, $C_{12}:C_2^3$ ($C_2^3$) x 7, $C_2^2\times D_{12}$ ($C_2^3$) x 7, $C_{12}:C_2^3$ ($C_2^3$) x 7

Order 64: $C_2^4:C_4$ ($D_6$) x 7, $C_2^3:D_4$ ($D_6$) x 7, $C_2^6$ ($D_6$)

Order 48: $C_2^2\times D_6$ ($C_2\times D_4$) x 168, $C_2^3\times C_6$ ($C_2\times D_4$) x 28, $C_2^2\times D_6$ ($C_2^4$) x 9, $C_2^3\times C_6$ ($C_2^4$) x 8, $C_2^2\times C_{12}$ ($C_2^4$) x 7, $C_6.C_2^3$ ($C_2^4$) x 7

Order 32: $C_2^3:C_4$ ($C_2\times D_6$) x 21, $C_2^5$ ($C_2\times D_6$) x 7, $C_2^2\times D_4$ ($C_2\times D_6$) x 7

Order 24: $C_2\times D_6$ ($C_2^2\wr C_2$) x 112, $C_2\times D_6$ ($C_2^2\times D_4$) x 56, $C_2^2\times C_6$ ($C_2^2\times D_4$) x 42, $C_2^2\times C_6$ ($C_2^5$)

Order 16: $C_2^4$ ($S_3\times D_4$) x 28, $C_2^4$ ($C_2^2\times D_6$) x 8, $C_2^2\times C_4$ ($C_2^2\times D_6$) x 7

Order 12: $D_6$ ($C_2^3:D_4$) x 56, $C_2\times C_6$ ($C_2^3:D_4$) x 28, $C_2\times C_6$ ($D_4\times C_2^3$) x 7

Order 8: $C_2^3$ ($D_4\times D_6$) x 42, $C_2^3$ ($C_2^3\times D_6$)

Order 6: $S_3$ ($C_2^3\wr C_2$) x 8, $C_6$ ($C_2^4:D_4$) x 7

Order 4: $C_2^2$ ($C_2^4:D_6$) x 28, $C_2^2$ ($C_{12}:C_2^4$) x 7

Order 3: $C_3$ ($C_2^5:D_4$)

Order 2: $C_2$ ($C_2^5:D_6$) x 7

Order 1: $C_1$ ($C_2^6:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2^6:D_6$ ($C_1$)

Order 384: $C_2^4:D_{12}$ ($C_2$), $C_2^5.D_6$ ($C_2$), $D_6\times C_2^5$ ($C_2$), $C_2^5:D_6$ ($C_2$), $C_2^6:C_6$ ($C_2$), $C_2^6:S_3$ ($C_2$), $C_2^5.D_6$ ($C_2$)

Order 192: $D_6\times C_2^4$ ($C_2^2$) x 2, $C_2^4:C_{12}$ ($C_2^2$), $C_2^4.D_6$ ($C_2^2$), $C_2^3.D_{12}$ ($C_2^2$), $C_2^5\times C_6$ ($C_2^2$), $C_{12}:C_2^4$ ($C_2^2$), $C_2^5:C_6$ ($C_2^2$), $C_2^4:D_6$ ($C_2^2$), $C_2^4:D_6$ ($C_2^2$), $C_2^3:D_{12}$ ($C_2^2$), $C_2^4.D_6$ ($C_2^2$)

Order 128: $C_2^3\wr C_2$ ($S_3$)

Order 96: $C_2^3\times D_6$ ($C_2^3$) x 3, $C_2^3:D_6$ ($C_2^3$) x 2, $C_2^4\times C_6$ ($C_2^3$), $C_2^3\times D_6$ ($D_4$), $C_{12}:C_2^3$ ($C_2^3$), $C_2^2\times D_{12}$ ($C_2^3$), $C_{12}:C_2^3$ ($C_2^3$), $C_2^3:C_{12}$ ($C_2^3$), $C_2^3.D_6$ ($C_2^3$), $C_2^2.D_{12}$ ($C_2^3$)

Order 64: $C_2^4:C_4$ ($D_6$), $C_2^6$ ($D_6$), $C_2^3:D_4$ ($D_6$)

Order 48: $C_2^2\times D_6$ ($C_2^4$) x 3, $C_2^3\times C_6$ ($C_2^4$) x 2, $C_2^2\times D_6$ ($C_2\times D_4$) x 2, $C_2^3\times C_6$ ($C_2\times D_4$), $C_2^2\times C_{12}$ ($C_2^4$), $C_6.C_2^3$ ($C_2^4$)

Order 32: $C_2^5$ ($C_2\times D_6$), $C_2^2\times D_4$ ($C_2\times D_6$), $C_2^3:C_4$ ($C_2\times D_6$)

Order 24: $C_2\times D_6$ ($C_2^2\times D_4$) x 2, $C_2^2\times C_6$ ($C_2^5$), $C_2^2\times C_6$ ($C_2^2\times D_4$), $C_2\times D_6$ ($C_2^2\wr C_2$)

Order 16: $C_2^4$ ($C_2^2\times D_6$) x 2, $C_2^4$ ($S_3\times D_4$), $C_2^2\times C_4$ ($C_2^2\times D_6$)

Order 12: $D_6$ ($C_2^3:D_4$) x 2, $C_2\times C_6$ ($D_4\times C_2^3$), $C_2\times C_6$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($C_2^3\times D_6$), $C_2^3$ ($D_4\times D_6$)

Order 6: $C_6$ ($C_2^4:D_4$), $S_3$ ($C_2^3\wr C_2$)

Order 4: $C_2^2$ ($C_{12}:C_2^4$), $C_2^2$ ($C_2^4:D_6$)

Order 3: $C_3$ ($C_2^5:D_4$)

Order 2: $C_2$ ($C_2^5:D_6$)

Order 1: $C_1$ ($C_2^6:D_6$)

Series

Derived series

$C_2^6:D_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_2^6:D_6$

$\rhd$

$C_2^5.D_6$

$\rhd$

$C_2^4:C_{12}$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6:D_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3\wr C_2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $132 \times 132$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.