C₂²×#38500 #72293723

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

Aliases

direct product

C₂²×#38500

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

83809 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598543
maximal normal subgroup of	C₂²×#8598543
maximal quotient of	C₂²×#647849, C₂²×#633924, C₂²×#2310627

GAP

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

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

C22×#38500 #72293723