Abstract group 3072.ec: $C_2^8:D_6$

• Label: 3072.ec
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2^8:D_6$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^9.\POPlus(4,3)$, of order \(294912\)\(\medspace = 2^{15} \cdot 3^{2} \)
Composition factors:	$C_2$ x 10, $C_3$
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6
Elements	1	367	512	1680	512	3072
Conjugacy classes	1	33	1	22	1	58
Divisions	1	33	1	22	1	58
Autjugacy classes	1	7	1	4	1	14

Dimension	1	2	3	6	12
Irr. complex chars.	4	2	12	26	14	58
Irr. rational chars.	4	2	12	26	14	58

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{6}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,11)(2,12)(3,9)(4,10)(5,16,6,15)(7,14,8,13), (1,9,3,10,2,12)(4,11)(5,15,6,14,8,13)(7,16), (1,16)(2,15)(3,13)(4,14)(5,12,6,11)(7,9,8,10)\rangle$
Transitive group:	16T1526	24T5340	24T6012	24T6021	all 6
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^8$ $\,\rtimes\,$ $D_6$	$(C_2^4:D_4)$ $\,\rtimes\,$ $S_4$	$(C_2^6:S_4)$ $\,\rtimes\,$ $C_2$	$(C_2^8:C_6)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $(C_2\times S_4)$	$C_2^4$ . $(C_2^3:S_4)$	$C_2^2$ . $(C_2^5:S_4)$		more information
Aut. group:	$\Aut(C_6^3.C_2^3)$

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 526804 subgroups in 42223 conjugacy classes, 25 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^8:D_6$
Commutator:	$G' \simeq$ $C_2^6:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\wr C_2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^8:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2^2$
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

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 3072: $C_2^8:D_6$

Order 1536: $C_2^6:S_4$ x 2, $C_2^8:C_6$

Order 1024: $C_2^8.C_2^2$

Order 768: $C_2^5:S_4$ x 3, $C_2^6:A_4$

Order 512: $C_2^6.D_4$ x 9, $C_2^7:C_2^2$ x 3, $C_2^4\wr C_2$ x 3

Order 384: $C_2^4:S_4$ x 18, $C_2^5:A_4$ x 4

Order 256: $C_2^6:C_4$ x 22, $C_2^5:D_4$ x 22, $C_2^4.C_2^4$ x 18, $C_2^5:D_4$ x 18, $C_2^5.D_4$ x 9, $C_2^4.C_4^2$ x 6, $C_2^8$, $C_4^2:C_2^4$

Order 192: $C_2^4:A_4$ x 30, $C_2^3:S_4$ x 6, $C_2^3:S_4$

Order 128: $C_2^3\wr C_2$ x 158, $C_2^5:C_4$ x 136, $C_2^4:D_4$ x 53, $C_2^4:D_4$ x 36, $C_2^4.D_4$ x 36, $C_2^4.D_4$ x 36, $C_2^7$ x 30, $C_2^4:Q_8$ x 12, $C_2^4:D_4$ x 12, $C_2^4.D_4$ x 9, $C_2^5:C_4$ x 9, $C_4^2:C_2^3$ x 9, $D_4^2:C_2$ x 9, $C_2^3.C_4^2$ x 6, $D_4^2:C_2$ x 6

Order 96: $C_2^2:S_4$ x 42, $C_2^3:A_4$ x 13, $C_2^3:A_4$

Order 64: $C_2^6$ x 1005, $C_2^3:D_4$ x 529, $C_2^4:C_4$ x 526, $C_2^4:C_4$ x 136, $C_2^4:C_4$ x 36, $C_2^2.C_4^2$ x 36, $D_4:D_4$ x 36, $C_2\wr C_2^2$ x 24, $D_4\times C_2^3$ x 22, $C_4^2:C_2^2$ x 18, $D_4^2$ x 18, $C_4^2:C_2^2$ x 18, $C_2^3:D_4$ x 18, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 6, $D_4:C_2^3$ x 3, $C_2^3:Q_8$ x 3, $C_4^2:C_2^2$ x 2

Order 48: $C_2^2:A_4$ x 112, $C_2\times S_4$ x 3

Order 32: $C_2^5$ x 8353, $C_2^3:C_4$ x 789, $C_2^2\wr C_2$ x 744, $C_2^2\times D_4$ x 281, $C_4:D_4$ x 54, $C_2^3:C_4$ x 36, $C_2^2.D_4$ x 36, $C_4\times D_4$ x 36, $C_2^3\times C_4$ x 22, $C_4^2:C_2$ x 18, $C_2^2:Q_8$ x 18, $D_4:C_2^2$ x 12, $C_4^2:C_2$ x 12, $D_4:C_2^2$ x 9, $C_4^2:C_2$ x 9, $C_4:D_4$ x 6

Order 24: $S_4$ x 18, $C_2\times A_4$ x 4

Order 16: $C_2^4$ x 17268, $C_2\times D_4$ x 553, $C_2^2:C_4$ x 544, $C_2^2\times C_4$ x 145, $C_4:C_4$ x 18, $D_4:C_2$ x 18, $C_4^2$ x 6, $C_2\times Q_8$ x 3

Order 12: $A_4$ x 30, $D_6$

Order 8: $C_2^3$ x 8406, $D_4$ x 167, $C_2\times C_4$ x 145, $Q_8$ x 3

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

Order 4: $C_2^2$ x 1028, $C_4$ x 22

Order 3: $C_3$

Order 2: $C_2$ x 33

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3072: $C_2^8:D_6$

Order 1536: $C_2^6:S_4$, $C_2^8:C_6$

Order 1024: $C_2^8.C_2^2$

Order 768: $C_2^5:S_4$, $C_2^6:A_4$

Order 512: $C_2^4\wr C_2$ x 2, $C_2^6.D_4$ x 2, $C_2^7:C_2^2$

Order 384: $C_2^4:S_4$ x 2, $C_2^5:A_4$ x 2

Order 256: $C_2^6:C_4$ x 4, $C_2^5:D_4$ x 4, $C_2^4.C_2^4$ x 3, $C_2^5:D_4$ x 3, $C_2^5.D_4$ x 2, $C_2^8$, $C_4^2:C_2^4$, $C_2^4.C_4^2$

Order 192: $C_2^4:A_4$ x 5, $C_2^3:S_4$, $C_2^3:S_4$

Order 128: $C_2^5:C_4$ x 13, $C_2^3\wr C_2$ x 11, $C_2^4:D_4$ x 9, $C_2^7$ x 5, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4:Q_8$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_4^2:C_2^3$ x 2, $D_4^2:C_2$ x 2, $C_2^4:D_4$ x 2, $C_2^3.C_4^2$, $D_4^2:C_2$

Order 96: $C_2^2:S_4$ x 4, $C_2^3:A_4$ x 3, $C_2^3:A_4$

Order 64: $C_2^6$ x 48, $C_2^3:D_4$ x 30, $C_2^4:C_4$ x 28, $C_2^4:C_4$ x 13, $C_2^4:C_4$ x 4, $D_4\times C_2^3$ x 4, $C_2^2.C_4^2$ x 4, $C_2^3:D_4$ x 4, $D_4:D_4$ x 3, $C_2\wr C_2^2$ x 3, $C_4^2:C_2^2$ x 2, $D_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $D_4:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3:Q_8$

Order 48: $C_2^2:A_4$ x 12, $C_2\times S_4$

Order 32: $C_2^5$ x 188, $C_2^2\wr C_2$ x 46, $C_2^3:C_4$ x 44, $C_2^2\times D_4$ x 22, $C_2^2.D_4$ x 5, $C_4:D_4$ x 5, $C_2^3:C_4$ x 4, $C_2^3\times C_4$ x 4, $C_4\times D_4$ x 3, $D_4:C_2^2$ x 2, $D_4:C_2^2$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4^2:C_2$ x 2, $C_4:D_4$, $C_4^2:C_2$

Order 24: $C_2\times A_4$ x 2, $S_4$ x 2

Order 16: $C_2^4$ x 362, $C_2\times D_4$ x 33, $C_2^2:C_4$ x 30, $C_2^2\times C_4$ x 15, $D_4:C_2$ x 3, $C_4:C_4$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $A_4$ x 5, $D_6$

Order 8: $C_2^3$ x 196, $C_2\times C_4$ x 15, $D_4$ x 13, $Q_8$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 53, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3072: $C_2^8:D_6$ ($C_1$)

Order 1536: $C_2^6:S_4$ ($C_2$) x 2, $C_2^8:C_6$ ($C_2$)

Order 768: $C_2^6:A_4$ ($C_2^2$)

Order 512: $C_2^4\wr C_2$ ($S_3$)

Order 256: $C_2^8$ ($D_6$)

Order 128: $C_2^4:D_4$ ($S_4$) x 3

Order 64: $C_2^6$ ($C_2\times S_4$) x 3

Order 32: $C_2^5$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($C_2^3:S_4$) x 6, $C_2^4$ ($C_2^3:S_4$)

Order 4: $C_2^2$ ($C_2^5:S_4$) x 3

Order 1: $C_1$ ($C_2^8:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3072: $C_2^8:D_6$ ($C_1$)

Order 1536: $C_2^6:S_4$ ($C_2$), $C_2^8:C_6$ ($C_2$)

Order 768: $C_2^6:A_4$ ($C_2^2$)

Order 512: $C_2^4\wr C_2$ ($S_3$)

Order 256: $C_2^8$ ($D_6$)

Order 128: $C_2^4:D_4$ ($S_4$)

Order 64: $C_2^6$ ($C_2\times S_4$)

Order 32: $C_2^5$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($C_2^3:S_4$), $C_2^4$ ($C_2^3:S_4$)

Order 4: $C_2^2$ ($C_2^5:S_4$)

Order 1: $C_1$ ($C_2^8:D_6$)

Series

Derived series

$C_2^8:D_6$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^8:D_6$

$\rhd$

$C_2^8:C_6$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^8:D_6$

$\rhd$

$C_2^6:A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 9 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 $58 \times 58$ rational character table. Alternatively, you may search for characters of this group with desired properties.