C₃²×#3417 #11770466

order	1152
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₃²×#3417

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×#41 ◁ C₃²×#131 ◁ G

1050 conjugacy classes of subgroups; only the 540 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 540 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) = 70
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 70 points:

(1,2,3)(7,8,13,18)(9,19,23,47)(10,15,24,38)(11,27,31,45)(12,32,29,17)(14,34,37,21)(16,41,43,30)(20,46,48,68)(22,51,33,42)(25,54,52,67)(26,57,53,40)(28,44,59,36)(35,60,61,49)(39,64,63,56)(50,69,58,65)(55,66,70,62);(4,5,6)(7,9,13,23)(8,14,18,37)(10,20,24,48)(11,28,31,59)(12,33,29,22)(15,35,38,61)(16,42,43,51)(17,44,32,36)(19,45,47,27)(21,41,34,30)(25,55,52,70)(26,58,53,50)(39,65,63,69)(40,66,57,62)(46,67,68,54)(49,64,60,56);(7,10)(8,15)(9,20)(11,25)(12,26)(13,24)(14,35)(16,39)(17,40)(18,38)(19,46)(21,49)(22,50)(23,48)(27,54)(28,55)(29,53)(30,56)(31,52)(32,57)(33,58)(34,60)(36,62)(37,61)(41,64)(42,65)(43,63)(44,66)(45,67)(47,68)(51,69)(59,70);(10,24)(15,38)(20,48)(25,52)(26,53)(35,61)(39,63)(40,57)(46,68)(49,60)(50,58)(54,67)(55,70)(56,64)(62,66)(65,69)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4477);