D₁₂₂₁₂ #72224059

order	24424
automorphism	Aut(G) order 71806560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₂₁₂

Splittings

Fitting	C₄×C₄₃×C₇₁ ⋊ C₂
Hall	C₄₃×C₇₁ ⋊ D₄
Sylow	C₇₁ ⋊ D₁₇₂
Sylow	C₄₃ ⋊ D₂₈₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₀₅₃
commutator, socle, cosocle	C₂×C₄₃×C₇₁	C₂²
Fitting	C₄×C₄₃×C₇₁	C₂
hypercenter	C₄	D₃₀₅₃
hypocenter	C₄₃×C₇₁	D₄
LCS	G ▷ C₂×C₄₃×C₇₁ ▷ C₄₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 12212

A minimal faithful permutation representation — 2 generators on 118 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);(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)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Polycyclic presentation:

2^3,43,71|b^2=c,b^a=bc,d^a=d^42,e^a=e^70

Conjugacy classes & character table

6109 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₄₂₄
maximal normal subgroup of	D₂₄₄₂₄
maximal quotient of	Dic₁₂₂₁₂, D₂₄₄₂₄

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, 71]);;
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.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12212 #72224059