Dic₁₅₃₇₀ #72293418

order	61480
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₃₇₀

Splittings

Hall	C₅×C₂₉×C₅₃ ⋊ Q₈
Hall	C₂₉×C₅₃ ⋊ Dic₁₀
Hall	C₅×C₅₃ ⋊ Dic₅₈
Hall	C₅×C₂₉ ⋊ Dic₁₀₆
Sylow	C₅₃ ⋊ Dic₂₉₀
Sylow	C₂₉ ⋊ Dic₅₃₀
Sylow	C₅ ⋊ Dic₃₀₇₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₆₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₅₃	C₂²
Fitting	C₄×C₅×C₂₉×C₅₃	C₂
hypercenter	C₄	D₇₆₈₅
hypocenter	C₅×C₂₉×C₅₃	Q₈
LCS	G ▷ C₂×C₅×C₂₉×C₅₃ ▷ C₅×C₂₉×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 61480

A minimal faithful permutation representation — 2 generators on 95 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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^3,5,29,53|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^28,f^a=f^52

Conjugacy classes & character table

15373 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, 5, 29, 53]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15370 #72293418