C₃⋊D₂₁₇₅ #27189388

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₂₁₇₅
unambiguous semidirect product	C₂₅⋊(C₃⋊D₈₇)
unambiguous semidirect product	C₂₉⋊(C₃⋊D₇₅)

Splittings

Fitting, derived, Hall	C₃²×C₂₅×C₂₉ ⋊ C₂
Hall	C₂₅×C₂₉ ⋊ C₃⋊D₃
Hall	C₃²×C₂₉ ⋊ D₂₅
Hall	C₃²×C₂₅ ⋊ D₂₉
Sylow	C₂₉ ⋊ C₃⋊D₇₅
Sylow	C₂₅ ⋊ C₃⋊D₈₇
Sylow	C₃² ⋊ D₇₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃²×C₂₅×C₂₉	C₂
Frattini	C₅	C₃⋊D₄₃₅
socle	C₃²×C₅×C₂₉	D₅
LCS	G ▷ C₃²×C₂₅×C₂₉
UCS	C₁

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 6525

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

Polycyclic presentation:

2,3^2,5,29,5|d^5=f^4,b^a=b^2,c^a=c^2,d^a=d^4f,e^a=e^28,f^a=f^4

Conjugacy classes & character table

3264 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉×C₃⋊D₇₅, D₂₅×C₃⋊D₈₇, C₃⋊D₃×D₇₂₅, …
maximal normal subgroup of	D₂₉×C₃⋊D₇₅, D₂₅×C₃⋊D₈₇, C₃⋊D₃×D₇₂₅, …
maximal quotient of	C₃⋊Dic₂₁₇₅, C₂×C₃⋊D₂₁₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 20);

C3⋊D2175 #27189388