C₂²×#39048 #72339442

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

Aliases

direct product

C₂²×#39048

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

79873 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598997
maximal normal subgroup of	C₂²×#8598997
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((5,6,9,12)(7,8,11,10)(21,22)(23,28)(24,26)(25,27), (3,4), (5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,23,25,24)(22,26,27,28), (1,2)(3,4), (13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,28)(25,27), (5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27), (5,8)(6,11)(7,12)(9,10)(13,14)(15,17)(16,19)(18,20)(21,24,25,23)(22,26,27,28));

C22×#39048 #72339442