C₂×#72225116 #72321182

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#72225116

Subgroups

	subgroup	G/H
center, socle	C₂²	#483875
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	#346	C₂⁵
prime radical	C₂×#513022	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×#11406 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3203 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)(5,16,19,40,21,41,18,43)(6,12)(7,26,29,60)(8,31)(9,23)(10,15)(11,20,35,47,36,52,55,49)(13,30,38,62)(14,33)(17,39,44,53,45,64,46,63)(22,51,50,42,54,65,48,66)(24,57,28,59)(25,32)(27,58,37,61)(34,56);(3,5,7,20,8,22,30,53,9,19,28,49,33,50,61,39,10,21,29,52,34,54,62,63,6,18,24,47,25,48,58,64)(4,11,13,16,14,17,26,42,12,35,27,43,31,44,57,51,15,36,38,41,32,45,60,66,23,55,37,40,56,46,59,65);(1,2);(3,4)(5,17,18,46,21,45,19,44)(6,23)(7,27,29,37)(8,32)(9,12)(10,15)(11,22,55,48,36,54,35,50)(13,28,38,24)(14,34)(16,39,43,63,41,64,40,53)(20,51,49,66,52,65,47,42)(25,31)(26,58,60,61)(30,59,62,57)(33,56);(3,6,10,9)(4,12,15,23)(5,18,21,19)(7,28,29,24)(8,25,34,33)(11,35,36,55)(13,37,38,27)(14,31,32,56)(16,40,41,43)(17,44,45,46)(20,49,52,47)(22,48,54,50)(26,59,60,57)(30,61,62,58)(39,63,64,53)(42,65,66,51)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225118
maximal normal subgroup of	C₂×#72225118

GAP

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

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

C2×#72225116 #72321182