C₂²×#10785 #72314751

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

Aliases

direct product

C₂²×#10785

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

47117 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)(10,14)(11,19)(12,21)(15,17)(16,22)(18,23);(6,8)(9,10,12,18,13,16,21,17)(11,14,20,15,19,22,24,23);(10,15,16,23)(12,21)(14,17,22,18)(20,24);(1,2)(10,16)(14,22)(15,23)(17,18);(3,4)(5,7)(6,8)(9,11,13,19)(10,14,16,22)(12,20,21,24)(15,17,23,18)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314759, #72314851, C₂×#72239632, …
maximal normal subgroup of	#72314759, #72314851, C₂×#72239632, …

GAP

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

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

C22×#10785 #72314751