C₂²×#7675391 #72278402

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

Aliases

direct product

C₂²×#7675391

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

At least 7587 conjugacy classes of subgroups; only the 473192 normal subgroups are shown.

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:

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

Conjugacy classes & character table

440 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×#72219693, C₂²×#72219674, C₂×#72323423

GAP

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

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

C22×#7675391 #72278402