C₂×#496097 #72286174

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#496097

Subgroups

	subgroup	G/H
center	C₂×C₁₆	#127
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	#21683
prime radical	C₂×#23794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 66 points:

(3,4)(5,11)(6,12)(7,26,27,55)(8,14)(9,16)(10,15)(13,29,40,45)(17,31)(18,47,48,56)(19,33)(20,35)(21,34)(22,36)(23,52,49,66)(24,38)(25,39)(28,57,58,65)(30,43)(32,50,41,53)(37,54,62,63)(42,51,61,64)(44,59)(46,60);(1,2)(13,40)(26,55)(32,41)(37,62)(42,61)(47,56)(52,66)(57,65);(3,5,8,19,9,20,22,44,10,21,30,6,17,24,46,25)(4,11,14,33,16,35,36,59,15,34,43,12,31,38,60,39)(7,18,28,49,29,50,51,63,27,48,58,23,45,53,64,54)(13,41,42,37,26,56,57,52,40,32,61,62,55,47,65,66);(3,6,22,5,17,44,8,24,10,19,46,21,9,25,30,20)(4,12,36,11,31,59,14,38,15,33,60,34,16,39,43,35)(7,23,51,18,45,63,28,53,27,49,64,48,29,54,58,50)(13,37,57,32,55,66,42,56,40,62,65,41,26,52,61,47);(3,7)(4,13,15,40)(5,18)(6,23)(8,28)(9,29)(10,27)(11,32,34,41)(12,37,33,62)(14,42,43,61)(16,26,31,55)(17,45)(19,49)(20,50)(21,48)(22,51)(24,53)(25,54)(30,58)(35,47,38,56)(36,57,60,65)(39,52,59,66)(44,63)(46,64)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224961
maximal normal subgroup of	C₂×#72224961
automorphism group of	C₄×C₆₄, #7510
outer automorphism group of	C₄×C₆₄

GAP

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

gap> G := Group((3,4)(5,11)(6,12)(7,26,27,55)(8,14)(9,16)(10,15)(13,29,40,45)(17,31)(18,47,48,56)(19,33)(20,35)(21,34)(22,36)(23,52,49,66)(24,38)(25,39)(28,57,58,65)(30,43)(32,50,41,53)(37,54,62,63)(42,51,61,64)(44,59)(46,60), (1,2)(13,40)(26,55)(32,41)(37,62)(42,61)(47,56)(52,66)(57,65), (3,5,8,19,9,20,22,44,10,21,30,6,17,24,46,25)(4,11,14,33,16,35,36,59,15,34,43,12,31,38,60,39)(7,18,28,49,29,50,51,63,27,48,58,23,45,53,64,54)(13,41,42,37,26,56,57,52,40,32,61,62,55,47,65,66), (3,6,22,5,17,44,8,24,10,19,46,21,9,25,30,20)(4,12,36,11,31,59,14,38,15,33,60,34,16,39,43,35)(7,23,51,18,45,63,28,53,27,49,64,48,29,54,58,50)(13,37,57,32,55,66,42,56,40,62,65,41,26,52,61,47), (3,7)(4,13,15,40)(5,18)(6,23)(8,28)(9,29)(10,27)(11,32,34,41)(12,37,33,62)(14,42,43,61)(16,26,31,55)(17,45)(19,49)(20,50)(21,48)(22,51)(24,53)(25,54)(30,58)(35,47,38,56)(36,57,60,65)(39,52,59,66)(44,63)(46,64));

C2×#496097 #72286174