C₂²×#15324 #72339455

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

Aliases

direct product

C₂²×#15324

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

63957 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#647923, C₂²×#3860035, C₂²×#1418518
maximal normal subgroup of	C₂²×#647923, C₂²×#3860035, C₂²×#1418518

GAP

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

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

C22×#15324 #72339455