C₂²×#36187 #72301162

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

Aliases

direct product

C₂²×#36187

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

87661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#7671954, C₂²×#7687902, C₂²×#7985575
maximal normal subgroup of	C₂²×#7671954, C₂²×#7687902, C₂²×#7985575

GAP

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

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

C22×#36187 #72301162