D₁₁₁₂₄ #72215831

order	22248
automorphism	Aut(G) order 40847328
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₂₄

Splittings

Fitting	C₄×C₂₇×C₁₀₃ ⋊ C₂
Hall	C₂₇×C₁₀₃ ⋊ D₄
Sylow	C₁₀₃ ⋊ D₁₀₈
Sylow	C₂₇ ⋊ D₄₁₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₇₈₁
commutator, cosocle	C₂×C₂₇×C₁₀₃	C₂²
Frattini	C₂×C₉	C₂×D₃₀₉
Fitting	C₄×C₂₇×C₁₀₃	C₂
socle	C₂×C₃×C₁₀₃	C₂×D₉
hypercenter	C₄	D₂₇₈₁
hypocenter	C₂₇×C₁₀₃	D₄
LCS	G ▷ C₂×C₂₇×C₁₀₃ ▷ C₂₇×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 11124

A minimal faithful permutation representation — 2 generators on 134 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134);(2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(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:

2^3,3,103,3^2|b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^102,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

5565 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₄₈
maximal normal subgroup of	D₂₂₂₄₈
maximal quotient of	Dic₁₁₁₂₄, D₂₂₂₄₈

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

D11124 #72215831