C₂²×#12574 #72236748

order	1024
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#12574

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#99
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

29245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#147567, C₂²×#146654, C₂²×#498927, …
maximal normal subgroup of	C₂²×#147567, C₂²×#146654, C₂²×#498927, …
maximal quotient of	#72236751

GAP

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

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

C22×#12574 #72236748