D₁₄₀₇₆ #72235656

order	28152
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₀₇₆

Splittings

Fitting	C₄×C₉×C₁₇×C₂₃ ⋊ C₂
Hall	C₉×C₁₇×C₂₃ ⋊ D₄
Hall	C₁₇×C₂₃ ⋊ D₃₆
Hall	C₉×C₂₃ ⋊ D₆₈
Hall	C₉×C₁₇ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₆₁₂
Sylow	C₁₇ ⋊ D₈₂₈
Sylow	C₉ ⋊ D₁₅₆₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₅₁₉
commutator, cosocle	C₂×C₉×C₁₇×C₂₃	C₂²
Frattini	C₂×C₃	C₂×D₁₁₇₃
Fitting	C₄×C₉×C₁₇×C₂₃	C₂
socle	C₂×C₃×C₁₇×C₂₃	C₂×D₃
hypercenter	C₄	D₃₅₁₉
hypocenter	C₉×C₁₇×C₂₃	D₄
LCS	G ▷ C₂×C₉×C₁₇×C₂₃ ▷ C₉×C₁₇×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 14076

A minimal faithful permutation representation — 2 generators on 53 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Polycyclic presentation:

2^3,3,17,23,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^16,f^a=f^22,g^a=g^2

Conjugacy classes & character table

7041 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₁₅₂
maximal normal subgroup of	D₂₈₁₅₂
maximal quotient of	Dic₁₄₀₇₆, D₂₈₁₅₂

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, 17, 23, 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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14076 #72235656