C₇×C₃⋊#13181892 #49138797

order	36414
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×C₃⋊#13181892
unambiguous semidirect product	C₃⋊(C₇×#13181892)

Splittings

Fitting, socle	C₃×C₇×C₁₇² ⋊ D₃
derived, Hall	C₃×C₁₇²⋊C₃ ⋊ C₂×C₇
Hall	C₃×C₇×C₁₇²⋊C₃ ⋊ C₂
Hall	C₇×C₁₇² ⋊ C₃⋊D₃
Sylow	C₁₇² ⋊ C₇×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₃⋊#13181892
commutator, cosocle, hypocenter	C₃×C₁₇²⋊C₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₇²	D₃
LCS	G ▷ C₃×C₁₇²⋊C₃
UCS	C₁ ◁ C₇

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 3 generators on 61 points:

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

Conjugacy classes & character table

1218 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 92);

C7×C3⋊#13181892 #49138797