C₂²×#15160 #72321582

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#15160

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75221 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#4306369, C₂²×#790170, C₂²×#1393449, …
maximal normal subgroup of	C₂²×#4306369, C₂²×#790170, C₂²×#1393449, …
maximal quotient of	C₂²×#154635, C₂²×#158362

GAP

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

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

C22×#15160 #72321582