C₂²×#15467 #72293786

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#15467

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

144173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2109240, C₂²×#4122215, C₂²×#845298, …
maximal normal subgroup of	C₂²×#2109240, C₂²×#4122215, C₂²×#845298, …

GAP

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

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

C22×#15467 #72293786