C₃⋊(C₅⋊D₄₄₅) #27211172

order	13350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product	C₃⋊(C₅⋊D₄₄₅)
unambiguous semidirect product	C₅⋊D₁₃₃₅
unambiguous semidirect product	C₈₉⋊(C₃⋊(C₅⋊D₅))

Splittings

Fitting, socle, derived, Hall	C₃×C₅²×C₈₉ ⋊ C₂
Hall	C₅²×C₈₉ ⋊ D₃
Hall	C₃×C₈₉ ⋊ C₅⋊D₅
Sylow	C₈₉ ⋊ C₃⋊(C₅⋊D₅)
Hall	C₃×C₅² ⋊ D₈₉
Sylow	C₅² ⋊ D₂₆₇
Sylow	C₃ ⋊ C₅⋊D₄₄₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₈₉	C₂
LCS	G ▷ C₃×C₅²×C₈₉
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 6675

A minimal faithful permutation representation — 3 generators on 102 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)(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)(91,92)(94,95)(96,97)(99,100)(101,102);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(90,91,92)(98,99,101,102,100);(93,94,96,97,95)

Polycyclic presentation:

2,3,5^2,89|b^a=b^2,c^a=c^4,d^a=d^4,e^a=e^88

Conjugacy classes & character table

3339 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅×D₁₃₃₅, D₁₅×D₄₄₅, C₃⋊(C₅⋊D₅)×D₈₉, …
maximal normal subgroup of	D₅×D₁₃₃₅, D₁₅×D₄₄₅, C₃⋊(C₅⋊D₅)×D₈₉, …
maximal quotient of	C₃⋊#22581820, C₂×C₃⋊(C₅⋊D₄₄₅)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 16);

C3⋊(C5⋊D445) #27211172