C₂²×#48144 #72301299

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

Aliases

direct product

C₂²×#48144

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

69357 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8757652
maximal normal subgroup of	C₂²×#8757652

GAP

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

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

C22×#48144 #72301299