C₂²×#35847 #72299838

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

Aliases

direct product

C₂²×#35847

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

94837 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7696044
maximal normal subgroup of	C₂²×#7696044

GAP

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

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

C22×#35847 #72299838