C₂²×#613192 #72339156

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#613192

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

1063640 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)(13,14,16,17)(15,18,19,20)(21,22,24,26)(23,25,27,28);(5,6)(7,8)(9,10,11,12)(14,17)(15,19)(21,23,24,27)(22,25,26,28);(6,8)(10,12)(23,27)(25,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,25)(24,27)(26,28);(3,4);(1,2)(3,4);(13,14)(15,20)(16,17)(18,19)(21,22)(23,28)(24,26)(25,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72225737
maximal normal subgroup of	C₂²×#72225737

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

C22×#613192 #72339156