C₃×C₇⋊D₃₇₁ #29510064

order	15582
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₃×C₇⋊D₃₇₁
unambiguous semidirect product	C₅₃⋊(C₃×C₇⋊D₇)

Splittings

Fitting, socle, Hall	C₃×C₇²×C₅₃ ⋊ C₂
derived, Hall	C₇²×C₅₃ ⋊ C₂×C₃
Hall	C₃×C₅₃ ⋊ C₇⋊D₇
Hall	C₃×C₇² ⋊ D₅₃
Sylow	C₅₃ ⋊ C₃×C₇⋊D₇
Sylow	C₇² ⋊ C₃×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₇⋊D₃₇₁
commutator, cosocle, hypocenter	C₇²×C₅₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₅₃	C₂
LCS	G ▷ C₇²×C₅₃
UCS	C₁ ◁ C₃

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 7791

A minimal faithful permutation representation — 3 generators on 70 points:

(1,2,3)(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)(58,59)(60,61)(62,63)(65,66)(67,68)(69,70);(4,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8,6)(64,65,67,69,70,68,66);(57,58,60,62,63,61,59)

Conjugacy classes & character table

3900 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#23780749, C₃×D₇×D₃₇₁, C₃×C₇⋊D₇×D₅₃, …
maximal normal subgroup of	C₃×#23780749, C₃×D₇×D₃₇₁, C₃×C₇⋊D₇×D₅₃, …
maximal quotient of	C₃×C₇⋊Dic₃₇₁, C₂×C₃×C₇⋊D₃₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15582, 42);

C3×C7⋊D371 #29510064