C₇×D₈₉₉ #27160407

order	12586
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₈₉₉
unambiguous semidirect product	C₃₁⋊(C₇×D₂₉)

Splittings

Fitting, socle, Hall	C₇×C₂₉×C₃₁ ⋊ C₂
derived, Hall	C₂₉×C₃₁ ⋊ C₂×C₇
Hall	C₇×C₃₁ ⋊ D₂₉
Hall	C₇×C₂₉ ⋊ D₃₁
Sylow	C₃₁ ⋊ C₇×D₂₉
Sylow	C₂₉ ⋊ C₇×D₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₈₉₉
commutator, cosocle, hypocenter	C₂₉×C₃₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₂₉×C₃₁	C₂
LCS	G ▷ C₂₉×C₃₁
UCS	C₁ ◁ C₇

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)(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)(32,33,35,37,39,41,43,45,47,49,51,53,55,57,59,60,58,56,54,52,50,48,46,44,42,40,38,36,34)

Conjugacy classes & character table

3157 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₂₉×D₃₁, D₇×D₈₉₉, D₂₀₃⋊D₃₁, …
maximal normal subgroup of	C₇×D₂₉×D₃₁, D₇×D₈₉₉, D₂₀₃⋊D₃₁, …
maximal quotient of	C₇×Dic₈₉₉, C₂×C₇×D₈₉₉, C₃×C₇×D₈₉₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12586, 5);

C7×D899 #27160407