D₂₁₈₁₂ #72275271

order	43624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₈₁₂

Splittings

Fitting	C₄×C₇×C₁₉×C₄₁ ⋊ C₂
Hall	C₇×C₁₉×C₄₁ ⋊ D₄
Hall	C₁₉×C₄₁ ⋊ D₂₈
Hall	C₇×C₄₁ ⋊ D₇₆
Hall	C₇×C₁₉ ⋊ D₁₆₄
Sylow	C₄₁ ⋊ D₅₃₂
Sylow	C₁₉ ⋊ D₁₁₄₈
Sylow	C₇ ⋊ D₃₁₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₄₅₃
commutator, socle, cosocle	C₂×C₇×C₁₉×C₄₁	C₂²
Fitting	C₄×C₇×C₁₉×C₄₁	C₂
hypercenter	C₄	D₅₄₅₃
hypocenter	C₇×C₁₉×C₄₁	D₄
LCS	G ▷ C₂×C₇×C₁₉×C₄₁ ▷ C₇×C₁₉×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 21812

A minimal faithful permutation representation — 2 generators on 71 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71);(2,4)(6,11)(7,10)(8,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^3,7,19,41|b^2=c,b^a=bc,d^a=d^6,e^a=e^18,f^a=f^40

Conjugacy classes & character table

10909 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 19, 41]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D21812 #72275271