C₂²×#54180 #72296212

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#54180

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

73913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811
maximal normal subgroup of	C₂²×#9228811
maximal quotient of	C₂²×#3825309

GAP

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

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

C22×#54180 #72296212