Dic₁₅₄₉₈ #72293673

order	61992
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₄₉₈

Splittings

Hall	C₇×C₂₇×C₄₁ ⋊ Q₈
Hall	C₂₇×C₄₁ ⋊ Dic₁₄
Hall	C₇×C₄₁ ⋊ Dic₅₄
Hall	C₇×C₂₇ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₃₇₈
Sylow	C₂₇ ⋊ Dic₅₇₄
Sylow	C₇ ⋊ Dic₂₂₁₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₇₄₉
commutator, cosocle	C₂×C₇×C₂₇×C₄₁	C₂²
Frattini	C₂×C₉	C₂×D₈₆₁
Fitting	C₄×C₇×C₂₇×C₄₁	C₂
socle, prime radical	C₂×C₃×C₇×C₄₁	C₂×D₉
hypercenter	C₄	D₇₇₄₉
hypocenter	C₇×C₂₇×C₄₁	Q₈
LCS	G ▷ C₂×C₇×C₂₇×C₄₁ ▷ C₇×C₂₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 61992

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4)(5,8,7,6)(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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,42)(38,41)(39,40)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

15501 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", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 41, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15498 #72293673