C₂²×#40494 #72326126

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#40494

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

111505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7656573, C₂²×#7966197, C₂²×#7645255
maximal normal subgroup of	C₂²×#7656573, C₂²×#7966197, C₂²×#7645255
maximal quotient of	C₂²×#1259290, C₂²×#773781, C₂²×#719640, …

GAP

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

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

C22×#40494 #72326126