C₂²×#14777 #72295709

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#14777

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

89533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#607767, C₂²×#846721, C₂²×#613868, …
maximal normal subgroup of	C₂²×#607767, C₂²×#846721, C₂²×#613868, …
maximal quotient of	C₂×#72334426

GAP

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

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

C22×#14777 #72295709