C₂²×#10740 #72314964

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10740

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

47805 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237174, C₂×#72239644, C₂²×#393975, …
maximal normal subgroup of	C₂×#72237174, C₂×#72239644, C₂²×#393975, …

GAP

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

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

C22×#10740 #72314964