D₃₀₂₆₄ #72295576

order	60528
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₂₆₄

Splittings

Fitting	C₃×C₈×C₁₃×C₉₇ ⋊ C₂
Hall	C₃×C₁₃×C₉₇ ⋊ D₈
Hall	C₁₃×C₉₇ ⋊ D₂₄
Hall	C₃×C₉₇ ⋊ D₁₀₄
Sylow	C₉₇ ⋊ D₃₁₂
Hall	C₃×C₁₃ ⋊ D₇₇₆
Sylow	C₁₃ ⋊ D₂₃₂₈
Sylow	C₃ ⋊ D₁₀₀₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₁₃₂
commutator, cosocle	C₃×C₄×C₁₃×C₉₇	C₂²
Frattini	C₄	C₂×D₃₇₈₃
Fitting	C₃×C₈×C₁₃×C₉₇	C₂
socle	C₂×C₃×C₁₃×C₉₇	D₄
hypercenter	C₈	D₃₇₈₃
hypocenter	C₃×C₁₃×C₉₇	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₉₇ ▷ C₂×C₃×C₁₃×C₉₇ ▷ C₃×C₁₃×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 30264

A minimal faithful permutation representation — 2 generators on 121 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);(2,8)(3,7)(4,6)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2^4,3,13,97|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^96

Conjugacy classes & character table

15135 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 13, 97]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30264 #72295576