C₂²×#37117 #72292899

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#37117

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

118901 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)(11,14)(12,13)(15,16)(18,19)(22,23);(3,4);(6,8)(10,12)(11,15)(17,18,21,19)(20,22,24,23);(11,15)(14,16);(5,6)(7,8)(9,11,13,15)(10,14,12,16)(17,19,21,18)(20,22,24,23);(1,2)(3,4);(9,12,13,10)(11,16,15,14)(17,20)(18,22)(19,23)(21,24)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7718113, C₂²×#8037281, C₂²×#7721725, …
maximal normal subgroup of	C₂²×#7718113, C₂²×#8037281, C₂²×#7721725, …
maximal quotient of	C₂²×#846721, C₂²×#763341, C₂²×#847053, …

GAP

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

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

C22×#37117 #72292899