C₂²×#42963 #72324860

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂²×#42963

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

102901 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)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,20,24,22);(6,8)(9,11,13,16)(10,14,12,15);(11,16)(14,15)(17,19,21,24)(18,22,23,20);(3,4);(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4);(5,6,7,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7783989
maximal normal subgroup of	C₂²×#7783989

GAP

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

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

C22×#42963 #72324860