C₂²×#7670204 #72321163

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂²×#7670204

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2081944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×D₄×#3484, C₂²×#72232065
maximal normal subgroup of	C₂²×D₄×#3484, C₂²×#72232065
maximal quotient of	C₂²×#120×#121, C₂²×#72233877

GAP

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

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

C22×#7670204 #72321163