C₂²×#12741 #72311407

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#12741

Subgroups

	subgroup	G/H
center, socle	C₂³	#1786
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

32229 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)(8,9)(10,23)(13,26)(14,16)(15,27)(17,33)(18,35)(19,22)(20,36)(24,25)(28,32)(29,34)(30,31);(1,2);(5,6,8,14,23,32,25,27,11,15,24,28,10,16,9,12)(7,17,18,31,36,30,19,33,21,13,22,29,20,34,35,26);(3,4);(5,7,11,21)(6,13,15,17)(8,22,24,18)(9,19,25,35)(10,20,23,36)(12,26,27,33)(14,31,28,29)(16,30,32,34)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500168, C₂²×#494058, C₂²×#499742, …
maximal normal subgroup of	C₂²×#500168, C₂²×#494058, C₂²×#499742, …
maximal quotient of	C₂×#72280536

GAP

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

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

C22×#12741 #72311407