C₂²×#153831 #72321968

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#153831

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁴×C₄²
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁸	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

770664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 6 generators on 24 points:

(3,4);(13,14,16,15)(18,20)(22,24);(9,10)(11,12)(14,15)(17,18)(19,20)(21,22,23,24);(6,8)(9,10,11,12)(13,15,16,14);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12)

Conjugacy classes & character table

608 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#95×#120, C₂²×#72278348, C₂²×#72278330
maximal normal subgroup of	C₂²×#95×#120, C₂²×#72278348, C₂²×#72278330

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4), (13,14,16,15)(18,20)(22,24), (9,10)(11,12)(14,15)(17,18)(19,20)(21,22,23,24), (6,8)(9,10,11,12)(13,15,16,14), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12));

C22×#153831 #72321968