C₂²×#4308012 #72334499

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂²×#4308012

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

971456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 28 points:

(5,6,7,8)(9,10,11,12)(13,14,17,18)(15,16,19,20)(21,22,24,26)(23,25,28,27);(6,8)(10,12)(15,19)(16,20)(21,23)(22,25)(24,28)(26,27);(5,6)(7,8)(14,18)(15,19);(9,10)(11,12)(13,15)(14,16)(17,19)(18,20);(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,27)(24,26)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233705
maximal normal subgroup of	C₂²×#72233705

GAP

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

gap> G := Group((5,6,7,8)(9,10,11,12)(13,14,17,18)(15,16,19,20)(21,22,24,26)(23,25,28,27), (6,8)(10,12)(15,19)(16,20)(21,23)(22,25)(24,28)(26,27), (5,6)(7,8)(14,18)(15,19), (9,10)(11,12)(13,15)(14,16)(17,19)(18,20), (5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,27)(24,26)(25,28), (3,4), (1,2)(3,4));

C22×#4308012 #72334499