D₇₈₇₅ #72212974

order	15750
automorphism	Aut(G) order 28350000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₇₈₇₅

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₂₅ ⋊ C₂
Hall	C₉×C₁₂₅ ⋊ D₇
Hall	C₇×C₁₂₅ ⋊ D₉
Sylow	C₁₂₅ ⋊ D₆₃
Hall	C₇×C₉ ⋊ D₁₂₅
Sylow	C₉ ⋊ D₈₇₅
Sylow	C₇ ⋊ D₁₁₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₂₅	C₂
Frattini	C₃×C₂₅	D₁₀₅
socle	C₃×C₅×C₇	D₇₅
LCS	G ▷ C₇×C₉×C₁₂₅
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 7875

A minimal faithful permutation representation — 2 generators on 141 points:

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

Polycyclic presentation:

2,3,5,7,3,5^2|b^3=e,c^5=f,f^5=g,b^a=b^2e^2,c^a=c^4f^4g^4,d^a=d^6,e^a=e^2,f^a=f^4g^4,g^a=g^4

Conjugacy classes & character table

3939 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₇₈₇₅, D₂₃₆₂₅
maximal normal subgroup of	C₂×D₇₈₇₅
maximal quotient of	Dic₇₈₇₅, C₂×D₇₈₇₅, D₂₃₆₂₅

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 3, 5, 7, 3, 5, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4*F.7^4);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.7^4);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7875 #72212974