C₂²×#54822 #72296629

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

Aliases

direct product

C₂²×#54822

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

75557 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)(7,10)(8,9)(11,12)(15,19)(18,20);(6,8)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(14,17)(15,19)(23,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,25,26,28);(5,8,9,6)(7,10,11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,26)(25,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8197496
maximal normal subgroup of	C₂²×#8197496
maximal quotient of	C₂²×#2171267

GAP

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

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

C22×#54822 #72296629