C₆₇×D₃₁⋊C₃ #27159576

order	12462
isoclinism	isoclinic class of D₃₁⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₇×D₃₁⋊C₃
unambiguous semidirect product	D₃₁⋊(C₃×C₆₇)
unambiguous semidirect product	(C₃₁⋊C₃)⋊(C₂×C₆₇)

Splittings

Fitting, socle, Hall	C₃₁×C₆₇ ⋊ C₂×C₃
derived, Sylow	C₃₁ ⋊ C₂×C₃×C₆₇
Hall	C₆₇×C₃₁⋊C₃ ⋊ C₂
Hall	D₃₁×C₆₇ ⋊ C₃
Hall	C₃₁⋊C₃ ⋊ C₂×C₆₇
Hall	D₃₁ ⋊ C₃×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	D₃₁⋊C₃
commutator, cosocle, hypocenter	C₃₁	C₂×C₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₃₁×C₆₇	C₂×C₃
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₆₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 2077

A minimal faithful permutation representation — 2 generators on 98 points:

(2,6,7,3,4,8)(5,14,15,9,10,22)(11,24,25,17,18,30)(12,23,19,20,16,13)(21,31,28,27,26,29)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(1,2,5,11,19,7,6,12,21,28,15,23,14,18,26,25,30,31,24,10,16,22,29,27,20,4,8,13,17,9,3)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₆₇×D₃₁⋊C₃, C₂×C₆₇×D₃₁⋊C₃, D₃₁⋊C₃×C₆₇⋊C₃, …
maximal normal subgroup of	D₆₇×D₃₁⋊C₃, C₂×C₆₇×D₃₁⋊C₃, D₃₁⋊C₃×C₆₇⋊C₃, …
maximal quotient of	C₆₇×(C₃₁⋊C₃)⋊C₄, C₂×C₆₇×D₃₁⋊C₃, C₆₇×D₃₁⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12462, 18);

C67×D31⋊C3 #27159576