Dic₁₄₈₅₈ #72292170

order	59432
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₅₈

Splittings

Hall	C₁₇×C₁₉×C₂₃ ⋊ Q₈
Hall	C₁₉×C₂₃ ⋊ Dic₃₄
Hall	C₁₇×C₂₃ ⋊ Dic₃₈
Hall	C₁₇×C₁₉ ⋊ Dic₄₆
Sylow	C₂₃ ⋊ Dic₆₄₆
Sylow	C₁₉ ⋊ Dic₇₈₂
Sylow	C₁₇ ⋊ Dic₈₇₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₂₉
commutator, socle, prime radical, cosocle	C₂×C₁₇×C₁₉×C₂₃	C₂²
Fitting	C₄×C₁₇×C₁₉×C₂₃	C₂
hypercenter	C₄	D₇₄₂₉
hypocenter	C₁₇×C₁₉×C₂₃	Q₈
LCS	G ▷ C₂×C₁₇×C₁₉×C₂₃ ▷ C₁₇×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 59432

A minimal faithful permutation representation — 2 generators on 67 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);(1,5,3,7)(2,6,4,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,17,19,23|a^2=c,b^2=c,b^a=bc,d^a=d^16,e^a=e^18,f^a=f^22

Conjugacy classes & character table

14861 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, 17, 19, 23]);;
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.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14858 #72292170