D₃×C₄₁²⋊C₃ #44475413

order	30258
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₄₁

Aliases

direct product	D₃×C₄₁²⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₄₁²⋊C₃)
unambiguous semidirect product	(D₃×C₄₁²)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₄₁² ⋊ C₂×C₃
Hall	C₃×C₄₁²⋊C₃ ⋊ C₂
Sylow	C₄₁² ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₁²	C₂×C₃
LCS	G ▷ C₃×C₄₁²
UCS	C₁

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 369

A minimal faithful permutation representation — 2 generators on 126 points:

(2,3)(4,5,7)(6,10,19)(8,14,29)(9,17,37)(11,22,48)(12,24,53)(13,27,58)(15,32,67)(16,35,72)(18,39,78)(20,43,62)(21,46,76)(23,51,31)(25,55,93)(26,57,47)(28,60,91)(30,63,95)(33,69,110)(34,42,59)(36,74,97)(38,77,84)(40,64,103)(41,54,96)(44,83,107)(45,86,113)(49,61,102)(50,68,108)(52,71,82)(56,99,122)(65,104,87)(66,105,120)(70,111,123)(73,114,117)(75,109,125)(79,94,118)(80,98,112)(81,100,92)(85,89,101)(88,116,119)(90,106,124)(115,121,126);(1,2,3)(4,6,11,23,52,94,99,123,114,37,76,108,29,62,102,55,98,121,78,57,100,72,113,60,67,107,124,103,24,54,97,104,27,59,101,105,110,77,116,125,63)(5,34,44,126,9,10,85,90,18,21,22,120,40,47,50,51,33,53,92,8,71,84,96,16,20,118,119,36,45,49,122,75,87,91,93,70,95,58,15,112,117)(7,111,61,35,12,66,19,73,25,86,41,69,48,17,80,28,74,38,31,46,115,32,65,88,82,68,39,83,13,109,79,14,26,106,42,30,56,43,81,64,89)

Conjugacy classes & character table

1689 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30258, 23);

D3×C412⋊C3 #44475413