C₂²×#10783 #72318647

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

Aliases

direct product

C₂²×#10783

Subgroups

	subgroup	G/H
center	C₂²×C₄	#386
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	C₂²⋊#95
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

39001 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279500
maximal normal subgroup of	C₂×#72279500

GAP

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

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

C22×#10783 #72318647