C₂×#72225021 #72322002

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#72225021

Subgroups

	subgroup	G/H
center	C₂×C₄	#11680
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×#346	C₂⁴
socle	C₂²	#136641
prime radical	C₂×#33600	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₄×C₈ ◁ C₂³×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 66 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225031, C₂×#72225074, C₂×#72225085
maximal normal subgroup of	C₂×#72225031, C₂×#72225074, C₂×#72225085

GAP

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

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

C2×#72225021 #72322002