Abstract group 12288.bck: $C_2^7.\GL(2,\mathbb{Z}/4)$

• Label: 12288.bck
• Order: \( 2^{12} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^7.\GL(2,\mathbb{Z}/4)$
Order:	\(12288\)\(\medspace = 2^{12} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^8.A_4.C_6.C_2^6.C_2$, of order \(2359296\)\(\medspace = 2^{18} \cdot 3^{2} \)
Composition factors:	$C_2$ x 12, $C_3$
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	1215	512	6976	1536	2048	12288
Conjugacy classes	1	125	1	74	3	4	208
Divisions	1	125	1	52	3	1	183
Autjugacy classes	1	28	1	8	3	1	42

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	10	24	0	110	56	208
Irr. rational chars.	4	6	12	3	88	70	183

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{6}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,4)(2,3)(5,12,6,11)(7,10,8,9)(13,21)(14,22)(15,24)(16,23), (1,15,5,17,11,24,2,16,6,18,12,23)(3,13,8,19,10,21,4,14,7,20,9,22)\rangle$
Transitive group:	24T11643				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{10}$ . $D_6$	$C_2^9$ . $D_{12}$	$(C_2^8.S_4)$ . $C_2$	$(C_2^8.S_4)$ . $C_2$	all 40

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 91 normal subgroups (26 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 $C_2^2$
Commutator:	a subgroup isomorphic to 1536.408544623
Frattini:	a subgroup isomorphic to $C_2^6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^3$

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 768) are shown, as well as all normal subgroups of any index.

Order 12288: $C_2^7.\GL(2,\mathbb{Z}/4)$

Order 6144: $C_2^8.S_4$, $C_2^8.S_4$, $C_2^9.C_{12}$

Order 4096: $C_2^9.C_2^3$

Order 3072: $C_2^7.S_4$ x 4, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^8.C_6.C_2$, $C_2^8.C_{12}$, $C_2^7.S_4$

Order 2048: $C_2^9.C_2^2$ x 9, $C_2^8.C_2^3$ x 4, $C_2^{10}.C_2$ x 2

Order 1536: $C_2^6:S_4$ x 9, $C_2^6.S_4$ x 9, $C_2^7:C_{12}$ x 4, $C_2^9.C_3$ x 3, $C_2^6:S_4$ x 2

Order 1024: $C_2^8.C_2^2$ x 89, $C_2^9.C_2$ x 52, $C_2^9.C_2$ x 47, $C_2^7.C_2^3$ x 41, $C_2^6.C_2^4$, $C_2^{10}$

Order 768: $C_2^5:S_4$ x 48, $C_2^6:A_4$ x 30, $C_2^5.S_4$ x 15, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^6:C_{12}$ x 5, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^6:A_4$

Order 512: $C_2^8.C_2$ x 3, $C_2^9$ x 3

Order 256: $C_2^8$ x 4

Order 128: $C_2^7$ x 9, $C_2^5\times C_4$

Order 64: $C_2^6$ x 10

Order 32: $C_2^5$ x 21

Order 16: $C_2^4$ x 10

Order 8: $C_2^3$ x 9

Order 4: $C_2^2$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 12288: $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 6144: $C_2^8.S_4$ ($C_2$), $C_2^8.S_4$ ($C_2$), $C_2^9.C_{12}$ ($C_2$)

Order 3072: $C_2^7.S_4$ ($C_4$) x 2, $C_2^8.C_6.C_2$ ($C_2^2$)

Order 2048: $C_2^{10}.C_2$ ($S_3$)

Order 1536: $C_2^9.C_3$ ($D_4$) x 2, $C_2^9.C_3$ ($C_2\times C_4$)

Order 1024: $C_2^{10}$ ($D_6$)

Order 768: $C_2^6:A_4$ ($C_2^2:C_4$)

Order 512: $C_2^8.C_2$ ($S_4$) x 3, $C_2^9$ ($C_3:D_4$), $C_2^9$ ($D_{12}$), $C_2^9$ ($C_4\times S_3$)

Order 256: $C_2^8$ ($C_2\times S_4$) x 3, $C_2^8$ ($D_6:C_4$)

Order 128: $C_2^7$ ($\GL(2,\mathbb{Z}/4)$) x 3, $C_2^7$ ($C_4:S_4$) x 3, $C_2^7$ ($C_4\times S_4$) x 3, $C_2^5\times C_4$ ($C_2^2:S_4$)

Order 64: $C_2^6$ ($C_2^3:S_4$) x 6, $C_2^6$ ($C_2.\GL(2,\mathbb{Z}/4)$) x 3, $C_2^6$ ($C_2^3:S_4$)

Order 32: $C_2^5$ ($C_2^4:S_4$) x 6, $C_2^5$ ($C_2^4:D_{12}$) x 6, $C_2^5$ ($C_2^5.D_6$) x 6, $C_2^5$ ($C_2\wr S_3$), $C_2^5$ ($C_2^4:D_{12}$), $C_2^5$ ($C_2^5.D_6$)

Order 16: $C_2^4$ ($C_2^3.\GL(2,\mathbb{Z}/4)$) x 6, $C_2^4$ ($C_2^5:S_4$) x 3, $C_2^4$ ($C_2^3.\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2^4:\GL(2,\mathbb{Z}/4)$) x 3, $C_2^3$ ($C_2^6:D_{12}$) x 3, $C_2^3$ ($C_2^7.D_6$) x 3

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

Order 2: $C_2$ ($C_2^6.\GL(2,\mathbb{Z}/4)$), $C_2$ ($C_2^9.D_6$), $C_2$ ($C_2^8.D_{12}$)

Order 1: $C_1$ ($C_2^7.\GL(2,\mathbb{Z}/4)$)

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 8 larger groups in the database.

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

Character theory

Complex character table

See the $208 \times 208$ 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 $183 \times 183$ rational character table (warning: may be slow to load).