D₁₁₇₀₀ #72223945

order	23400
automorphism	Aut(G) order 33696000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₀₀

Splittings

Fitting	C₄×C₉×C₁₃×C₂₅ ⋊ C₂
Hall	C₉×C₁₃×C₂₅ ⋊ D₄
Hall	C₁₃×C₂₅ ⋊ D₃₆
Hall	C₉×C₂₅ ⋊ D₅₂
Hall	C₉×C₁₃ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₄₆₈
Sylow	C₁₃ ⋊ D₉₀₀
Sylow	C₉ ⋊ D₁₃₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₉₂₅
commutator, cosocle	C₂×C₉×C₁₃×C₂₅	C₂²
Frattini	C₂×C₃×C₅	C₂×D₁₉₅
Fitting	C₄×C₉×C₁₃×C₂₅	C₂
socle	C₂×C₃×C₅×C₁₃	C₂×D₁₅
hypercenter	C₄	D₂₉₂₅
hypocenter	C₉×C₁₃×C₂₅	D₄
LCS	G ▷ C₂×C₉×C₁₃×C₂₅ ▷ C₉×C₁₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 11700

A minimal faithful permutation representation — 2 generators on 51 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^3,3,5,13,3,5|b^2=c,d^3=g,e^5=h,b^a=bc,d^a=d^2g^2,e^a=e^4h^4,f^a=f^12,g^a=g^2,h^a=h^4

Conjugacy classes & character table

5853 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₄₀₀
maximal normal subgroup of	D₂₃₄₀₀
maximal quotient of	Dic₁₁₇₀₀, D₂₃₄₀₀

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 13, 3, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
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.5^4*F.8^4);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);