C₂²×#45457 #72325883

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

Aliases

direct product

C₂²×#45457

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

162813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8108057, C₂²×#8598997
maximal normal subgroup of	C₂²×#8108057, C₂²×#8598997
maximal quotient of	C₂²×#851167

GAP

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

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

C22×#45457 #72325883