C₂²×#36725 #72296613

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

Aliases

direct product

C₂²×#36725

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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8197496, C₂²×#8108057, C₂²×#8387575
maximal normal subgroup of	C₂²×#8197496, C₂²×#8108057, C₂²×#8387575
maximal quotient of	C₂²×#2032000, C₂²×#851167

GAP

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

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

C22×#36725 #72296613