D₂₈₁₇₅ #72291505

order	56350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₁₇₅

Splittings

Fitting, derived, Hall	C₂₃×C₂₅×C₄₉ ⋊ C₂
Hall	C₂₅×C₄₉ ⋊ D₂₃
Hall	C₂₃×C₄₉ ⋊ D₂₅
Hall	C₂₃×C₂₅ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₅₇₅
Sylow	C₂₅ ⋊ D₁₁₂₇
Sylow	C₂₃ ⋊ D₁₂₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₃×C₂₅×C₄₉	C₂
Frattini	C₅×C₇	D₈₀₅
socle	C₅×C₇×C₂₃	D₃₅
LCS	G ▷ C₂₃×C₂₅×C₄₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 28175

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

Polycyclic presentation:

2,5,7,23,5,7|b^5=e,c^7=f,b^a=b^4e^4,c^a=c^6f^6,d^a=d^22,e^a=e^4,f^a=f^6

Conjugacy classes & character table

14089 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 5, 7, 23, 5, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.6^6);;
gap> SetConjugate(coll, 4, 1, F.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28175 #72291505