C₃²×#2195 #11769244

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2195

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#131 ◁ G

618 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 3 generators on 70 points:

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3255);