C₂²×#154077 #72291312

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#154077

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁴×C₄²
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

8307144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72240113, C₂²×#72286393, C₂²×#95×#120
maximal normal subgroup of	C₂²×#72240113, C₂²×#72286393, C₂²×#95×#120

GAP

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

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

C22×#154077 #72291312