C₂²×#618316 #72339144

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

Aliases

direct product

C₂²×#618316

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

1129928 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:

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

Conjugacy classes & character table

368 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((6,8)(9,10)(11,12)(13,14,16,18)(15,19,20,17)(21,22,24,25)(23,27,28,26), (10,12)(23,28)(26,27), (5,6,7,8)(9,10)(11,12)(15,20)(17,19)(22,25)(26,27), (13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,28)(25,27), (3,4), (1,2)(3,4), (13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28));

C22×#618316 #72339144