D₃₉₅₆ #72211277

order	7912
automorphism	Aut(G) order 7310688
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₉₅₆

Splittings

Fitting	C₄×C₂₃×C₄₃ ⋊ C₂
Hall	C₂₃×C₄₃ ⋊ D₄
Sylow	C₄₃ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₁₇₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₉₈₉
commutator, socle, cosocle	C₂×C₂₃×C₄₃	C₂²
Fitting	C₄×C₂₃×C₄₃	C₂
hypercenter	C₄	D₉₈₉
hypocenter	C₂₃×C₄₃	D₄
LCS	G ▷ C₂×C₂₃×C₄₃ ▷ C₂₃×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 3956

A minimal faithful permutation representation — 2 generators on 70 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);(2,4)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,23,43|b^2=c,b^a=bc,d^a=d^22,e^a=e^42

Conjugacy classes & character table

1981 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₉₁₂, D₁₁₈₆₈, D₁₉₇₈₀, …
maximal normal subgroup of	D₇₉₁₂
maximal quotient of	D₇₉₁₂, Dic₃₉₅₆, D₁₁₈₆₈, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 23, 43]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3956 #72211277