C₃⋊D₂₁₉₃ #27190488

order	13158
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₂₁₉₃
unambiguous semidirect product	C₁₇⋊(C₃⋊D₁₂₉)
unambiguous semidirect product	C₄₃⋊(C₃⋊D₅₁)

Splittings

Fitting, socle, derived, Hall	C₃²×C₁₇×C₄₃ ⋊ C₂
Hall	C₁₇×C₄₃ ⋊ C₃⋊D₃
Hall	C₃²×C₄₃ ⋊ D₁₇
Hall	C₃²×C₁₇ ⋊ D₄₃
Sylow	C₄₃ ⋊ C₃⋊D₅₁
Sylow	C₁₇ ⋊ C₃⋊D₁₂₉
Sylow	C₃² ⋊ D₇₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₇×C₄₃	C₂
LCS	G ▷ C₃²×C₁₇×C₄₃
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 6579

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

Polycyclic presentation:

2,3^2,17,43|b^a=b^2,c^a=c^2,d^a=d^16,e^a=e^42

Conjugacy classes & character table

3291 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×D₂₁₉₃, D₅₁×D₁₂₉, D₄₃×C₃⋊D₅₁, …
maximal normal subgroup of	D₃×D₂₁₉₃, D₅₁×D₁₂₉, D₄₃×C₃⋊D₅₁, …
maximal quotient of	C₃⋊Dic₂₁₉₃, C₂×C₃⋊D₂₁₉₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13158, 22);

C3⋊D2193 #27190488