C₃₁×D₆₇⋊C₃ #27159571

order	12462
isoclinism	isoclinic class of D₆₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₁×D₆₇⋊C₃
unambiguous semidirect product	D₆₇⋊(C₃×C₃₁)
unambiguous semidirect product	(C₆₇⋊C₃)⋊(C₂×C₃₁)

Splittings

Fitting, socle, Hall	C₃₁×C₆₇ ⋊ C₂×C₃
derived, Sylow	C₆₇ ⋊ C₂×C₃×C₃₁
Hall	C₃₁×C₆₇⋊C₃ ⋊ C₂
Hall	C₃₁×D₆₇ ⋊ C₃
Hall	C₆₇⋊C₃ ⋊ C₂×C₃₁
Hall	D₆₇ ⋊ C₃×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃₁	D₆₇⋊C₃
commutator, cosocle, hypocenter	C₆₇	C₂×C₃×C₃₁
Frattini	C₁	G
Fitting, socle	C₃₁×C₆₇	C₂×C₃
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₃₁

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

Conjugacy classes & character table

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

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12462, 13);

C31×D67⋊C3 #27159571