C₁₇×C₃⋊D₁₂₉ #27190490

order	13158
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₁₇×C₃⋊D₁₂₉
unambiguous semidirect product	C₃⋊(C₁₇×D₁₂₉)
unambiguous semidirect product	C₄₃⋊(C₁₇×C₃⋊D₃)

Splittings

Fitting, socle, Hall	C₃²×C₁₇×C₄₃ ⋊ C₂
derived, Hall	C₃²×C₄₃ ⋊ C₂×C₁₇
Hall	C₁₇×C₄₃ ⋊ C₃⋊D₃
Hall	C₃²×C₁₇ ⋊ D₄₃
Sylow	C₄₃ ⋊ C₁₇×C₃⋊D₃
Sylow	C₃² ⋊ C₁₇×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₇	C₃⋊D₁₂₉
commutator, cosocle, hypocenter	C₃²×C₄₃	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₁₇×C₄₃	C₂
LCS	G ▷ C₃²×C₄₃
UCS	C₁ ◁ C₁₇

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)(48,49)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(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)(47,48,49);(44,45,46)

Conjugacy classes & character table

3315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₁₇×D₁₂₉, D₅₁⋊D₁₂₉, C₁₇×C₃⋊D₃×D₄₃, …
maximal normal subgroup of	D₃×C₁₇×D₁₂₉, D₅₁⋊D₁₂₉, C₁₇×C₃⋊D₃×D₄₃, …
maximal quotient of	C₁₇×C₃⋊Dic₁₂₉, C₂×C₁₇×C₃⋊D₁₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13158, 24);

C17×C3⋊D129 #27190490