C₂²×#37782 #72324645

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

Aliases

direct product

C₂²×#37782

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

110965 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7694192, C₂²×#7783989, C₂²×#7722013, …
maximal normal subgroup of	C₂²×#7694192, C₂²×#7783989, C₂²×#7722013, …

GAP

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

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

C22×#37782 #72324645