Abstract group 3072.bjf: $C_2\wr D_6.C_4$

• Label: 3072.bjf
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 8
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $16$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2\wr D_6.C_4$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6\times C_2^4:C_3.C_2^4$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
Composition factors:	$C_2$ x 10, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	287	32	1760	352	640	3072
Conjugacy classes	1	35	1	96	7	12	152
Divisions	1	35	1	63	7	7	114
Autjugacy classes	1	23	1	36	5	5	71

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	32	24	32	4	56	0	4	0	152
Irr. rational chars.	16	20	16	8	36	1	16	1	114

Minimal presentations

Permutation degree:	$16$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

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:	${\langle a, b, c, d, e, f, g, h \mid a^{4}=b^{2}=c^{6}=d^{4}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,16)(2,5)(3,8)(9,14), (1,11)(2,12)(3,8)(5,15)(6,16)(9,14), (2,5)(9,14) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2\wr D_6)$ . $C_4$ (8)	$C_4$ . $(C_2\wr D_6)$ (2)	$(C_2^5.D_6)$ . $D_4$ (4)	$(C_2^5:D_4)$ . $D_6$	all 118
Aut. group:	$\Aut(C_{10}.C_2^4)$	$\Aut(C_5\times Q_8:\GL(2,3))$

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 214 normal subgroups (120 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2\times C_4$
Commutator:	a subgroup isomorphic to $C_2^3:A_4$
Frattini:	a subgroup isomorphic to $C_2^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 16 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 3072: $C_2\wr D_6.C_4$

Order 1536: $C_2^7.D_6$, $C_2^7.D_6$, $C_2^7:D_6$, $C_4\times C_2^4:S_4$, $C_2^6.C_2.D_6$, $C_4\times C_2^4:C_3.D_4$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2^2.S_3$, $C_2^6.C_2.D_6$, $C_2^7.D_6$, $C_4\times C_2^4.D_{12}$, $C_4\times C_2^4.(C_4\times S_3)$, $C_2^4.C_2^5.C_3$

Order 1024: $C_2^5.C_2^5$

Order 768: $C_2^6.D_6$ x 9, $C_2\wr D_6$ x 8, $C_4\times C_2^3:S_4$ x 4, $C_2^6.D_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_3.C_4^2$ x 2, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^2\times C_2^2:(A_4:C_4)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^2:(A_4:C_4)$, $C_2^5:S_4$, $C_4\times D_4\times S_4$, $C_2^5:D_{12}$, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^4.C_{12}:C_4$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^6:C_{12}$, $C_2^7:C_6$, $C_4\times C_2^4:A_4$, $C_2^5.(C_2\times C_{12})$, $C_2^6.C_2^2.C_3$, $C_2^6:C_{12}$, $C_2^4.C_2^4.C_3$

Order 512: $C_2^5.C_2^4$ x 4, $C_2^5.C_2^4$ x 4, $C_2^4.C_2^5$ x 2, $C_2^4.C_2^5$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_4^3:C_2^3$, $C_2^3.C_2^6$, $C_2^6:D_4$, $C_2^5.C_2^4$, $C_2^5.C_2^4$, $C_2^5.C_2^4$, $C_2^5.C_2^4$

Order 384: $C_2^5.D_6$ x 12, $C_2^2\wr S_3$ x 11, $C_2^5.D_6$ x 10, $C_2^4:S_4$ x 8, $C_2^4.S_4$ x 7, $C_2\wr C_6$ x 4, $C_2^4:S_4$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5:C_{12}$ x 4, $C_2\wr S_3$ x 4, $C_2^5:C_{12}$ x 3, $C_2^5.D_6$ x 3, $C_4\times \GL(2,\mathbb{Z}/4)$ x 3, $C_4\times C_2^3:A_4$ x 2, $C_2^4.S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_4$ x 2, $C_2^5.D_6$ x 2, $C_2^5:A_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5:A_4$, $C_4\times D_4\times A_4$, $C_2^5.D_6$, $C_4:(C_4\times S_4)$, $C_4:C_4\times S_4$, $C_4^2:S_4$, $C_4^2\times S_4$

Order 256: $C_2^3.C_2^5$ x 37, $C_2^5:D_4$ x 16, $C_2^4.C_2^4$ x 16, $C_2^5.D_4$ x 10, $C_2^5.D_4$ x 8, $C_2^5.D_4$ x 8, $C_2^4:C_4^2$ x 8, $C_2^5.D_4$ x 8, $C_2^5.D_4$ x 8, $C_2^4:C_4^2$ x 8, $C_2^5.D_4$ x 6, $C_2^5.C_2^3$ x 4, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^4:C_4^2$ x 4, $C_2^4:C_4^2$ x 4, $C_2^5.C_2^3$ x 4, $C_2^5.C_2^3$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^2.C_4^3$ x 4, $D_4^2:C_4$ x 4, $C_2^3.C_2^5$ x 4, $C_4^3:C_2^2$ x 4, $C_4\times D_4^2$ x 4, $C_2^5:D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^3.C_2^5$ x 3, $C_2^3.C_2^5$ x 3, $C_2^3.C_2^5$ x 3, $C_2^6:C_4$ x 2, $C_4^2:C_2^4$ x 2, $C_2^5:D_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^4:C_4^2$ x 2, $C_2^5.D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^6\times C_4$, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_4^2.C_2^4$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_4^3:C_2^2$, $C_4^3:C_2^2$, $C_2^6:C_4$, $C_4^3:C_2^2$

Order 192: $C_2^4.D_6$ x 30, $C_2^3:S_4$ x 22, $C_2^3:S_4$ x 8, $D_4\times S_4$ x 8, $C_2^3.S_4$ x 7, $C_2.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^2\wr C_3$ x 5, $A_4:C_4^2$ x 4, $C_2^4:C_{12}$ x 4, $C_2^4:C_{12}$ x 4, $C_2^4:A_4$ x 4, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^3\times S_4$ x 3, $C_2^4:C_{12}$ x 3, $C_2\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^3.S_4$ x 3, $C_2^4:C_{12}$ x 2, $A_4\times C_4:C_4$, $A_4\times C_4^2$, $C_2^3.D_{12}$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_4^2:D_6$

Order 128: $C_2^3\wr C_2$ x 4, $C_2^5:C_4$ x 3, $C_2^5\times C_4$ x 2, $C_2^7$, $C_2^4:D_4$, $C_4^2:C_2^3$

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

Order 64: $C_2^4\times C_4$ x 4, $C_2^6$ x 3, $C_2^4:C_4$ x 2, $C_2^3:D_4$ x 2, $C_2^4:C_4$ x 2, $D_4\times C_2^3$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 48: $C_2^2:A_4$

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

Order 16: $C_2^4$ x 6, $C_2^2\times C_4$ x 5

Order 8: $C_2^3$ x 4, $C_2\times C_4$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 3072: $C_2\wr D_6.C_4$ ($C_1$)

Order 1536: $C_2^7.D_6$ ($C_2$), $C_2^7.D_6$ ($C_2$), $C_2^7:D_6$ ($C_2$), $C_4\times C_2^4:S_4$ ($C_2$), $C_2^6.C_2.D_6$ ($C_2$), $C_4\times C_2^4:C_3.D_4$ ($C_2$), $C_2^6.C_2^2.S_3$ ($C_2$), $C_2^6.C_2^2.S_3$ ($C_2$), $C_2^6.C_2^2.S_3$ ($C_2$), $C_2^6.C_2^2.S_3$ ($C_2$), $C_2^6.C_2.D_6$ ($C_2$), $C_2^7.D_6$ ($C_2$), $C_4\times C_2^4.D_{12}$ ($C_2$), $C_4\times C_2^4.(C_4\times S_3)$ ($C_2$), $C_2^4.C_2^5.C_3$ ($C_2$)

Order 768: $C_2\wr D_6$ ($C_4$) x 8, $C_2^6.D_6$ ($C_2^2$) x 5, $C_2^6.D_6$ ($C_2^2$) x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 3, $C_2^4:C_3.C_4^2$ ($C_2^2$) x 2, $C_2^5:S_4$ ($C_2^2$), $C_2^3:\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^2\times C_2^2:(A_4:C_4)$ ($C_2^2$), $C_2^3.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^4.(C_6.Q_8)$ ($C_2^2$), $C_2^3.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_4\times C_2^2:(A_4:C_4)$ ($C_2^2$), $C_2^5:S_4$ ($C_2^2$), $C_2^5:D_{12}$ ($C_2^2$), $C_2^5:S_4$ ($C_2^2$), $C_2^5.S_4$ ($C_2^2$), $C_2^4.C_{12}:C_4$ ($C_2^2$), $C_2^4.(C_6.Q_8)$ ($C_2^2$), $C_2^3.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^6:C_{12}$ ($C_2^2$), $C_2^7:C_6$ ($C_2^2$), $C_4\times C_2^4:A_4$ ($C_2^2$), $C_2^5.(C_2\times C_{12})$ ($C_2^2$), $C_2^6.C_2^2.C_3$ ($C_2^2$), $C_2^6:C_{12}$ ($C_2^2$), $C_2^4.C_2^4.C_3$ ($C_2^2$)

Order 512: $C_2^4.C_2^5$ ($S_3$)

Order 384: $C_2^4.S_4$ ($C_2^3$) x 5, $C_2\wr C_6$ ($C_2\times C_4$) x 4, $C_2^4:S_4$ ($C_2\times C_4$) x 4, $C_2^4:D_{12}$ ($C_2\times C_4$) x 4, $C_2^5.D_6$ ($C_2\times C_4$) x 4, $C_2^2\wr S_3$ ($C_2\times C_4$) x 4, $C_2\wr S_3$ ($C_2\times C_4$) x 4, $C_2^5.D_6$ ($D_4$) x 4, $C_2^4:S_4$ ($C_2\times C_4$) x 4, $C_2^5:C_{12}$ ($C_2^3$) x 3, $C_2^2\wr S_3$ ($C_2^3$) x 3, $C_2^5:C_{12}$ ($C_2^3$) x 2, $C_2^5:A_4$ ($C_2^3$), $C_2^5:A_4$ ($C_2^3$)

Order 256: $C_2^5.D_4$ ($D_6$), $C_2^5.D_4$ ($D_6$), $C_2^6:C_4$ ($D_6$), $C_2^4:C_4^2$ ($D_6$), $C_2^6\times C_4$ ($D_6$), $C_2^5:D_4$ ($D_6$), $C_2^3.C_2^5$ ($D_6$)

Order 192: $C_2^3:S_4$ ($C_2^2\times C_4$) x 6, $C_2^4:C_{12}$ ($C_2^2\times C_4$) x 2, $C_2^2\wr C_3$ ($C_2^2\times C_4$) x 2, $C_2^3:S_4$ ($D_4:C_2$) x 2, $C_2^3:S_4$ ($C_2\times D_4$) x 2, $C_2^4:C_{12}$ ($C_2\times D_4$) x 2, $C_2^3.S_4$ ($C_2\times D_4$) x 2, $C_2^3.S_4$ ($C_2^2\times C_4$) x 2, $C_2^4:A_4$ ($C_2^2\times C_4$) x 2, $C_2^2\wr C_3$ ($C_2^4$)

Order 128: $C_2^3\wr C_2$ ($C_4\times S_3$) x 4, $C_2^5:C_4$ ($C_2\times D_6$) x 3, $C_2^5\times C_4$ ($C_2\times D_6$) x 2, $C_2^7$ ($C_2\times D_6$), $C_2^4:D_4$ ($C_2\times D_6$), $C_4^2:C_2^3$ ($S_4$)

Order 96: $C_2^2:S_4$ ($C_4\times D_4$) x 4, $C_2^3:A_4$ ($D_4:C_2^2$), $C_2^3:A_4$ ($C_2^2\times D_4$), $C_2^3:A_4$ ($C_2^3\times C_4$)

Order 64: $C_2^4:C_4$ ($C_4\times D_6$) x 2, $C_2^6$ ($C_4\times D_6$) x 2, $C_2^4\times C_4$ ($C_2\times S_4$) x 2, $C_2^4\times C_4$ ($S_3\times D_4$) x 2, $C_2^3:D_4$ ($C_4\times D_6$) x 2, $C_2^4:C_4$ ($C_2\times S_4$) x 2, $C_2^6$ ($C_2^2\times D_6$), $D_4\times C_2^3$ ($C_2\times S_4$), $C_2^3.D_4$ ($C_2\times S_4$), $C_2^2\times C_4^2$ ($C_2\times S_4$)

Order 48: $C_2^2:A_4$ ($C_4^2:C_2^2$)

Order 32: $C_2^3\times C_4$ ($C_2^2\times S_4$) x 5, $C_2^2\times D_4$ ($C_4\times S_4$) x 4, $C_2^5$ ($C_2^2\times S_4$) x 2, $C_2^5$ ($D_4:D_6$), $C_2^5$ ($D_4\times D_6$), $C_2^5$ ($C_{12}:C_2^3$)

Order 16: $C_2^4$ ($C_2^4.D_6$) x 4, $C_2^2\times C_4$ ($D_4\times S_4$) x 2, $C_2^2\times C_4$ ($C_2^4.D_6$) x 2, $C_2^4$ ($C_2^3\times S_4$), $C_2^4$ ($C_4^2:D_6$), $C_2^2\times C_4$ ($C_2^3:S_4$)

Order 8: $C_2\times C_4$ ($C_2^4:S_4$) x 2, $C_2^3$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2^3$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2^3$ ($C_2^5.D_6$), $C_2^3$ ($C_2^4:S_4$)

Order 4: $C_2^2$ ($C_4\times C_2^3:S_4$) x 2, $C_4$ ($C_2\wr D_6$) x 2, $C_2^2$ ($C_2^5:S_4$), $C_2^2$ ($C_4\times D_4\times S_4$)

Order 2: $C_2$ ($C_2\wr D_6:C_2$), $C_2$ ($C_2^7:D_6$), $C_2$ ($C_4\times C_2^4:S_4$)

Order 1: $C_1$ ($C_2\wr D_6.C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $152 \times 152$ 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 $114 \times 114$ rational character table.