D₃×C₇⋊(C₁₇²⋊C₃) #49138804

order	36414
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁₇

Aliases

direct product	D₃×C₇⋊(C₁₇²⋊C₃)
unambiguous semidirect product	C₃⋊(C₂×C₇⋊(C₁₇²⋊C₃))
unambiguous semidirect product	(D₃×C₇)⋊(C₁₇²⋊C₃)
unambiguous semidirect product	(D₃×C₁₇²)⋊ΓL(1,𝔽₈)

Splittings

Fitting, socle, derived	C₃×C₇×C₁₇² ⋊ C₂×C₃
Hall	C₃×C₇⋊(C₁₇²⋊C₃) ⋊ C₂
Hall	C₇×C₁₇² ⋊ C₃×D₃
Sylow	C₁₇² ⋊ D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃×C₁₇²⋊C₃

Subgroups

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

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 61 points:

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

Conjugacy classes & character table

2031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 99);

D3×C7⋊(C172⋊C3) #49138804