C₇×C₂₉×D₃₁ #27160405

order	12586
isoclinism	isoclinic class of D₃₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₂₉×D₃₁
unambiguous semidirect product	C₃₁⋊(C₂×C₇×C₂₉)

Splittings

Fitting, socle, Hall	C₇×C₂₉×C₃₁ ⋊ C₂
derived, Sylow	C₃₁ ⋊ C₂×C₇×C₂₉
Hall	C₂₉×C₃₁ ⋊ C₂×C₇
Hall	C₇×C₃₁ ⋊ C₂×C₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₂₉	D₃₁
commutator, cosocle, hypocenter	C₃₁	C₂×C₇×C₂₉
Frattini	C₁	G
Fitting, socle	C₇×C₂₉×C₃₁	C₂
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₇×C₂₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 6293

A minimal faithful permutation representation — 2 generators on 67 points:

(2,3)(4,5)(6,7)(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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)

Conjugacy classes & character table

3451 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₂₉×D₃₁, D₇×C₂₉×D₃₁, D₃₁×D₂₀₃, …
maximal normal subgroup of	C₇×D₂₉×D₃₁, D₇×C₂₉×D₃₁, D₃₁×D₂₀₃, …
maximal quotient of	C₇×C₂₉×Dic₃₁, C₂×C₇×C₂₉×D₃₁, C₃×C₇×C₂₉×D₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12586, 3);

C7×C29×D31 #27160405