C₃²×D₆₇ #11925860

order	1206
isoclinism	isoclinic class of D₆₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×D₆₇

Splittings

Fitting, socle, Hall	C₃²×C₆₇ ⋊ C₂
derived, Sylow	C₆₇ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₆₇
commutator, cosocle, hypocenter	C₆₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₆₇	C₂
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₃²

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 603

A minimal faithful permutation representation — 2 generators on 73 points:

(1,2,3)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(4,5,6)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Conjugacy classes & character table

315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₃×D₆₇, C₃⋊D₃×D₆₇, C₂×C₃²×D₆₇, …
maximal normal subgroup of	C₃×D₃×D₆₇, C₃⋊D₃×D₆₇, C₂×C₃²×D₆₇, …
maximal quotient of	C₃²×Dic₆₇, C₂×C₃²×D₆₇, C₃²×D₇₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1206, 11);

C32×D67 #11925860