C₂²×#59384 #72326111

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

Aliases

direct product

C₂²×#59384

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

83037 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8387575
maximal normal subgroup of	C₂²×#8387575
maximal quotient of	C₂²×#647923

GAP

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

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

C22×#59384 #72326111