C₂²×#15181 #72291889

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

Aliases

direct product

C₂²×#15181

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

80173 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#772802, C₂²×#647855, C₂²×#845354, …
maximal normal subgroup of	C₂²×#772802, C₂²×#647855, C₂²×#845354, …
maximal quotient of	C₂×#72298546

GAP

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

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

C22×#15181 #72291889