C₂×#72225075 #72319452

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#72225075

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₁₆	C₂⁴×C₈
Frattini, cosocle	#346	C₂⁵
prime radical	C₂²×#2263	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×#11405 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3768 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:

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

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225087, C₂×#72225103, C₂×#72225085
maximal normal subgroup of	C₂×#72225087, C₂×#72225103, C₂×#72225085
maximal quotient of	#72322903, #72323084

GAP

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

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

C2×#72225075 #72319452