C₂²×#3868090 #72297087

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#3868090

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

646608 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)(14,16)(18,20);(10,12)(13,14)(15,18)(16,17)(19,20);(6,8)(9,10)(11,12)(14,16)(18,20)(21,22,24,26)(23,25,27,28);(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,6,7,8)(9,10)(11,12)(13,16,17,14)(15,18,19,20)(21,23,24,27)(22,25,26,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72278822
maximal normal subgroup of	C₂×#72278822

GAP

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

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

C22×#3868090 #72297087