C₂²×#38366 #72236675

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#38366

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

105741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7792761, C₂³×#38366
maximal normal subgroup of	C₂²×#7792761, C₂³×#38366
maximal quotient of	C₂²×#619101, C₂³×#38366, C₂×#72232346, …

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), (9,10,13,14)(11,12,15,16)(17,18,21,20)(19,22,24,23), (6,8)(10,14)(11,15)(17,18,21,20)(19,23,24,22), (5,6)(7,8)(17,19)(18,22)(20,23)(21,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24), (9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,22,24,23));

C22×#38366 #72236675