C₇×D₉₄₉ #27210783

order	13286
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₉₄₉
unambiguous semidirect product	C₁₃⋊(C₇×D₇₃)
unambiguous semidirect product	C₇₃⋊(C₇×D₁₃)

Splittings

Fitting, socle, Hall	C₇×C₁₃×C₇₃ ⋊ C₂
derived, Hall	C₁₃×C₇₃ ⋊ C₂×C₇
Hall	C₇×C₇₃ ⋊ D₁₃
Hall	C₇×C₁₃ ⋊ D₇₃
Sylow	C₇₃ ⋊ C₇×D₁₃
Sylow	C₁₃ ⋊ C₇×D₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₉₄₉
commutator, cosocle, hypocenter	C₁₃×C₇₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₇₃	C₂
LCS	G ▷ C₁₃×C₇₃
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6643

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

Conjugacy classes & character table

3332 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×#16838291, C₇×#16838292, C₇×D₁₃×D₇₃, …
maximal normal subgroup of	C₇×#16838291, C₇×#16838292, C₇×D₁₃×D₇₃, …
maximal quotient of	C₇×Dic₉₄₉, C₂×C₇×D₉₄₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13286, 4);

C7×D949 #27210783