C₂²×#14064 #72330053

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#14064

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80477 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:

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

Conjugacy classes & character table

304 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#647938, C₂²×#1082743, C₂²×#515667
maximal normal subgroup of	C₂²×#647938, C₂²×#1082743, C₂²×#515667

GAP

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

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

C22×#14064 #72330053