C₂²×#12828 #72310990

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12828

Subgroups

	subgroup	G/H
center, socle	C₂³	#1804
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(6,12,14,28)(8,22)(9,24)(10,26)(13,27,31,33)(15,29,23,16)(17,32,34,36)(18,19)(20,35)(25,30);(1,2);(5,6,9,12,26,29,24,15,11,14,22,28,10,16,8,23)(7,17,25,36,35,13,30,27,21,34,19,32,20,31,18,33);(3,4);(1,2)(3,4)(5,7,11,21)(6,13)(8,18,24,30)(9,25,22,19)(10,20,26,35)(12,27)(14,31)(15,32)(16,17)(23,36)(28,33)(29,34)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500650, C₂²×#494058, C₂²×#500473
maximal normal subgroup of	C₂²×#500650, C₂²×#494058, C₂²×#500473

GAP

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

gap> G := Group((6,12,14,28)(8,22)(9,24)(10,26)(13,27,31,33)(15,29,23,16)(17,32,34,36)(18,19)(20,35)(25,30), (1,2), (5,6,9,12,26,29,24,15,11,14,22,28,10,16,8,23)(7,17,25,36,35,13,30,27,21,34,19,32,20,31,18,33), (3,4), (1,2)(3,4)(5,7,11,21)(6,13)(8,18,24,30)(9,25,22,19)(10,20,26,35)(12,27)(14,31)(15,32)(16,17)(23,36)(28,33)(29,34));

C22×#12828 #72310990