D₇₈₁₂ #72212935

order	15624
automorphism	Aut(G) order 16873920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₈₁₂

Splittings

Fitting	C₄×C₇×C₉×C₃₁ ⋊ C₂
Hall	C₇×C₉×C₃₁ ⋊ D₄
Hall	C₉×C₃₁ ⋊ D₂₈
Hall	C₇×C₃₁ ⋊ D₃₆
Hall	C₇×C₉ ⋊ D₁₂₄
Sylow	C₃₁ ⋊ D₂₅₂
Sylow	C₉ ⋊ D₈₆₈
Sylow	C₇ ⋊ D₁₁₁₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₉₅₃
commutator, cosocle	C₂×C₇×C₉×C₃₁	C₂²
Frattini	C₂×C₃	C₂×D₆₅₁
Fitting	C₄×C₇×C₉×C₃₁	C₂
socle	C₂×C₃×C₇×C₃₁	C₂×D₃
hypercenter	C₄	D₁₉₅₃
hypocenter	C₇×C₉×C₃₁	D₄
LCS	G ▷ C₂×C₇×C₉×C₃₁ ▷ C₇×C₉×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 7812

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)(15,20)(16,19)(17,18)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^3,3,7,31,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^6,f^a=f^30,g^a=g^2

Conjugacy classes & character table

3909 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₅₆₂₄, D₂₃₄₃₆
maximal normal subgroup of	D₁₅₆₂₄
maximal quotient of	Dic₇₈₁₂, D₁₅₆₂₄, D₂₃₄₃₆

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, 31, 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.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7812 #72212935