C₂²×#14407 #72339471

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14407

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

64493 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)(17,18)(19,23)(20,21)(22,24);(3,4);(5,6)(7,8)(9,10,11,12)(13,14)(15,16)(18,21)(19,24);(1,2)(3,4);(6,7)(17,18)(19,22)(20,21)(23,24);(5,7,8,6)(17,19)(18,22)(20,24)(21,23)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#873665, C₂²×#540671, C₂²×#1083597
maximal normal subgroup of	C₂²×#873665, C₂²×#540671, C₂²×#1083597

GAP

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

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

C22×#14407 #72339471