C₂²×#45645 #72291791

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂²×#45645

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

80609 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)(15,18)(16,20)(17,19)(21,22)(23,27)(24,25)(26,28);(5,6,8,10)(7,9,11,12);(7,11)(9,12)(13,15,17,16)(14,18,19,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,12)(8,10)(9,11)(13,16,17,15)(14,18,19,20)(21,24,25,22)(23,27,28,26);(5,7)(6,9)(8,11)(10,12)(13,14)(15,20)(16,18)(17,19)(21,23)(22,27)(24,26)(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×#8676694
maximal normal subgroup of	C₂²×#8676694
maximal quotient of	C₂²×#845307

GAP

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

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

C22×#45645 #72291791