D₁₁₅₁₅ #37527832

order	23030
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₁₅
unambiguous semidirect product	C₅⋊D₂₃₀₃
unambiguous semidirect product	C₄₇⋊D₂₄₅
unambiguous semidirect product	C₄₉⋊D₂₃₅

Splittings

Fitting, derived, Hall	C₅×C₄₇×C₄₉ ⋊ C₂
Hall	C₄₇×C₄₉ ⋊ D₅
Hall	C₅×C₄₉ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₂₄₅
Sylow	C₅ ⋊ D₂₃₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₄₇×C₄₉	C₂
Frattini	C₇	D₁₆₄₅
socle	C₅×C₇×C₄₇	D₇
LCS	G ▷ C₅×C₄₇×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 11515

A minimal faithful permutation representation — 2 generators on 101 points:

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

Polycyclic presentation:

2,5,7,47,7|c^7=e^6,b^a=b^4,c^a=c^6e,d^a=d^46,e^a=e^6

Conjugacy classes & character table

5759 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23030, 8);

D11515 #37527832