C₂²×#52315 #72296604

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

Aliases

direct product

C₂²×#52315

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

94769 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)(15,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,26);(7,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,24)(25,28)(26,27);(6,9)(7,11)(13,14,16,18)(15,19,20,17)(21,22,25,26)(23,27,28,24);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,24)(22,23)(25,27)(26,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8181078, C₂×#72232299, C₂×#72232314
maximal normal subgroup of	C₂²×#8181078, C₂×#72232299, C₂×#72232314
maximal quotient of	C₂²×#5473828, C₂²×#3332795, C₂²×#3333823

GAP

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

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

C22×#52315 #72296604