C₂²×#14338 #72296084

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

Aliases

direct product

C₂²×#14338

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

170805 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#4307938, C₂²×#521311, C₂²×#521312, …
maximal normal subgroup of	C₂²×#4307938, C₂²×#521311, C₂²×#521312, …
maximal quotient of	C₂²×#154095

GAP

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

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

C22×#14338 #72296084