C₂²×#59220 #72301653

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#59220

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

64253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8195808, C₂²×#7667013, C₂³×#59220
maximal normal subgroup of	C₂²×#8195808, C₂²×#7667013, C₂³×#59220
maximal quotient of	C₂³×#59220

GAP

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

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

C22×#59220 #72301653