D₁₁₈₄₄ #72223979

order	23688
automorphism	Aut(G) order 39227328
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₄₄

Splittings

Fitting	C₄×C₇×C₉×C₄₇ ⋊ C₂
Hall	C₇×C₉×C₄₇ ⋊ D₄
Hall	C₉×C₄₇ ⋊ D₂₈
Hall	C₇×C₄₇ ⋊ D₃₆
Hall	C₇×C₉ ⋊ D₁₈₈
Sylow	C₄₇ ⋊ D₂₅₂
Sylow	C₉ ⋊ D₁₃₁₆
Sylow	C₇ ⋊ D₁₆₉₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₉₆₁
commutator, cosocle	C₂×C₇×C₉×C₄₇	C₂²
Frattini	C₂×C₃	C₂×D₉₈₇
Fitting	C₄×C₇×C₉×C₄₇	C₂
socle	C₂×C₃×C₇×C₄₇	C₂×D₃
hypercenter	C₄	D₂₉₆₁
hypocenter	C₇×C₉×C₄₇	D₄
LCS	G ▷ C₂×C₇×C₉×C₄₇ ▷ C₇×C₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 11844

A minimal faithful permutation representation — 2 generators on 67 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,20)(16,19)(17,18)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^3,3,7,47,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^6,f^a=f^46,g^a=g^2

Conjugacy classes & character table

5925 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₈₈
maximal normal subgroup of	D₂₃₆₈₈
maximal quotient of	Dic₁₁₈₄₄, D₂₃₆₈₈

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, 7, 47, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11844 #72223979