D₁₁₅₃₆ #72216014

order	23072
automorphism	Aut(G) order 56480256
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₅₃₆

Splittings

Fitting	C₇×C₁₆×C₁₀₃ ⋊ C₂
Hall	C₇×C₁₀₃ ⋊ D₁₆
Sylow	C₁₀₃ ⋊ D₁₁₂
Sylow	C₇ ⋊ D₁₆₄₈

Subgroups

	subgroup	G/H
center	C₂	D₅₇₆₈
commutator, cosocle	C₇×C₈×C₁₀₃	C₂²
Frattini	C₈	C₂×D₇₂₁
Fitting	C₇×C₁₆×C₁₀₃	C₂
socle	C₂×C₇×C₁₀₃	D₈
hypercenter	C₁₆	D₇₂₁
hypocenter	C₇×C₁₀₃	D₁₆
LCS	G ▷ C₇×C₈×C₁₀₃ ▷ C₄×C₇×C₁₀₃ ▷ C₂×C₇×C₁₀₃ ▷ C₇×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 11536

A minimal faithful permutation representation — 2 generators on 126 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^5,7,103|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^102

Conjugacy classes & character table

5771 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₀₇₂
maximal normal subgroup of	D₂₃₀₇₂
maximal quotient of	Dic₁₁₅₃₆, D₂₃₀₇₂

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, 2, 7, 103]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11536 #72216014