Dic₁₃₈₂₅ #72290102

order	55300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₂₅

Splittings

derived, Hall	C₇×C₂₅×C₇₉ ⋊ C₄
Hall	C₂₅×C₇₉ ⋊ Dic₇
Hall	C₇×C₇₉ ⋊ Dic₂₅
Hall	C₇×C₂₅ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₇₅
Sylow	C₂₅ ⋊ Dic₅₅₃
Sylow	C₇ ⋊ Dic₁₉₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₈₂₅
commutator, hypocenter	C₇×C₂₅×C₇₉	C₄
Frattini	C₂×C₅	D₂₇₆₅
Fitting, cosocle	C₂×C₇×C₂₅×C₇₉	C₂
socle, prime radical	C₂×C₅×C₇×C₇₉	D₅
LCS	G ▷ C₇×C₂₅×C₇₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 55300

A minimal faithful permutation representation — 2 generators on 115 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,36)(32,35)(33,34)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^2,5,7,79,5|a^2=b,c^5=f,c^a=c^4f^4,d^a=d^6,e^a=e^78,f^a=f^4

Conjugacy classes & character table

13828 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, 5, 7, 79, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^78);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13825 #72290102