D₄₅×C₁₄₉ #27211776

order	13410
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₄₅×C₁₄₉
unambiguous semidirect product	C₅⋊(D₉×C₁₄₉)
unambiguous semidirect product	C₉⋊(D₅×C₁₄₉)

Splittings

Fitting, Hall	C₅×C₉×C₁₄₉ ⋊ C₂
derived, Hall	C₅×C₉ ⋊ C₂×C₁₄₉
Hall	C₉×C₁₄₉ ⋊ D₅
Hall	C₅×C₁₄₉ ⋊ D₉
Sylow	C₉ ⋊ D₅×C₁₄₉
Sylow	C₅ ⋊ D₉×C₁₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₄₉	D₄₅
commutator, cosocle, hypocenter	C₅×C₉	C₂×C₁₄₉
Frattini	C₃	D₁₅×C₁₄₉
Fitting	C₅×C₉×C₁₄₉	C₂
socle	C₃×C₅×C₁₄₉	D₃
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₁₄₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 163
transitive degree	μ_t(G) = 6705

A minimal faithful permutation representation — 2 generators on 163 points:

(2,3)(4,5)(7,9)(8,12)(10,11)(13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2,4,5,3)(6,14,9,12,11,10,8,7,13)

Conjugacy classes & character table

3576 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₅×D₁₄₉, D₅×D₉×C₁₄₉, D₄₅⋊D₁₄₉, …
maximal normal subgroup of	D₄₅×D₁₄₉, D₅×D₉×C₁₄₉, D₄₅⋊D₁₄₉, …
maximal quotient of	C₁₄₉×Dic₄₅, C₂×D₄₅×C₁₄₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13410, 7);

D45×C149 #27211776