D₂₇₄₉₂ #72290792

order	54984
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₄₉₂

Splittings

Fitting	C₃×C₄×C₂₉×C₇₉ ⋊ C₂
Hall	C₃×C₂₉×C₇₉ ⋊ D₄
Hall	C₂₉×C₇₉ ⋊ D₁₂
Hall	C₃×C₇₉ ⋊ D₁₁₆
Hall	C₃×C₂₉ ⋊ D₃₁₆
Sylow	C₇₉ ⋊ D₃₄₈
Sylow	C₂₉ ⋊ D₉₄₈
Sylow	C₃ ⋊ D₉₁₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₇₃
commutator, socle, cosocle	C₂×C₃×C₂₉×C₇₉	C₂²
Fitting	C₃×C₄×C₂₉×C₇₉	C₂
hypercenter	C₄	D₆₈₇₃
hypocenter	C₃×C₂₉×C₇₉	D₄
LCS	G ▷ C₂×C₃×C₂₉×C₇₉ ▷ C₃×C₂₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 27492

A minimal faithful permutation representation — 2 generators on 115 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);(2,4)(6,7)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^3,3,29,79|b^2=c,b^a=bc,d^a=d^2,e^a=e^28,f^a=f^78

Conjugacy classes & character table

13749 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, 2, 2, 3, 29, 79]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27492 #72290792