D₁₁₅₂₄ #72223910

order	23048
automorphism	Aut(G) order 63889056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₅₂₄

Splittings

Fitting	C₄×C₄₃×C₆₇ ⋊ C₂
Hall	C₄₃×C₆₇ ⋊ D₄
Sylow	C₆₇ ⋊ D₁₇₂
Sylow	C₄₃ ⋊ D₂₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₈₈₁
commutator, socle, cosocle	C₂×C₄₃×C₆₇	C₂²
Fitting	C₄×C₄₃×C₆₇	C₂
hypercenter	C₄	D₂₈₈₁
hypocenter	C₄₃×C₆₇	D₄
LCS	G ▷ C₂×C₄₃×C₆₇ ▷ C₄₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 114
transitive degree	μ_t(G) = 11524

A minimal faithful permutation representation — 2 generators on 114 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);(2,4)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^3,43,67|b^2=c,b^a=bc,d^a=d^42,e^a=e^66

Conjugacy classes & character table

5765 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₀₄₈
maximal normal subgroup of	D₂₃₀₄₈
maximal quotient of	Dic₁₁₅₂₄, D₂₃₀₄₈

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 43, 67]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11524 #72223910