C₂×#3417 #62634

order	256
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#3417

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1011 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 479 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3417⋊C₃
maximal subgroup of	#152170, #152172, #152175, …
maximal normal subgroup of	#152170, #152172, #152175, …
maximal quotient of	C₂×#33382, C₂×#33410, C₂×#33455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55622);