Abstract group 4096.brz: $C_2^9.C_2^3$

• Label: 4096.brz
• Order: \( 2^{12} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \)
• $\card{Z(G)}$: 16
• $\card{\Aut(G)}$: \( 2^{34} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{26} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $8$

Group information

Description:	$C_2^9.C_2^3$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	Group of order \(17179869184\)\(\medspace = 2^{34} \)
Composition factors:	$C_2$ x 12
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.

Group statistics

Order	1	2	4
Elements	1	1151	2944	4096
Conjugacy classes	1	233	262	496
Divisions	1	233	262	496

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$8$
Inequivalent generating 8-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{4}=d^{4}=e^{2}=f^{4}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $32$ $\langle(6,8)(14,16)(22,24)(30,31), (1,12)(4,9)(6,16)(8,14)(17,28)(20,25)(22,31) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_2^9.C_2^2)$ . $C_2$ (255)	$C_2$ . $(C_2^8.C_2^3)$ (15)	$C_2^2$ . $(C_2^8.C_2^2)$ (91)		more information

Elements of the group are displayed as permutations of degree 32.

Homology

Abelianization:	$C_{2}^{8} $
Schur multiplier:	$C_{2}^{25}$
Commutator length:	$1$

Subgroups

There are 448043 normal subgroups (41 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 $C_2^4$
Commutator:	a subgroup isomorphic to $C_2^4$
Frattini:	a subgroup isomorphic to $C_2^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^3$

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 2 (order at least 2048) are shown, as well as normal subgroups of index up to 2 or of order up to 4.

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

Order 2048: $C_2^7.C_2^4$ x 141, $C_2^8.C_2^3$ x 110, $C_2^9.C_2^2$ x 4

Order 4: $C_2^2$ x 91

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 2 or order up to 4 are shown.

Order 4096: $C_2^9.C_2^3$ ($C_1$)

Order 2048: $C_2^7.C_2^4$ ($C_2$) x 141, $C_2^8.C_2^3$ ($C_2$) x 110, $C_2^9.C_2^2$ ($C_2$) x 4

Order 1024: ? (unidentified group of order 4) x 10795

Order 512: ? (unidentified group of order 8) x 97219

Order 256: ? (unidentified group of order 16) x 202803

Order 128: ? (unidentified group of order 32) x 107731

Order 64: ? (unidentified group of order 64) x 23155

Order 32: ? (unidentified group of order 128) x 4531

Order 16: ? (unidentified group of order 256) x 1107

Order 8: ? (unidentified group of order 512) x 339

Order 4: $C_2^2$ ($C_2^7.C_2^3$) x 54, $C_2^2$ ($C_2^6.C_2^4$) x 22, $C_2^2$ ($C_2^8.C_2^2$) x 8, $C_2^2$ ($C_2^5.C_2^5$) x 7

Order 2: $C_2$ ($C_2^8.C_2^3$) x 10, $C_2$ ($C_2^7.C_2^4$) x 5

Order 1: $C_1$ ($C_2^9.C_2^3$)

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

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

The $496 \times 496$ rational character table is not available for this group.