Dic₁₃₄₄₇ #72289340

order	53788
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₄₄₇

Splittings

derived, Hall	C₇×C₁₇×C₁₁₃ ⋊ C₄
Hall	C₁₇×C₁₁₃ ⋊ Dic₇
Hall	C₇×C₁₁₃ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₁₁₃
Sylow	C₁₁₃ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₇₉₁
Sylow	C₇ ⋊ Dic₁₉₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₄₄₇
commutator, hypocenter	C₇×C₁₇×C₁₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₁₁₃	C₂
LCS	G ▷ C₇×C₁₇×C₁₁₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 53788

A minimal faithful permutation representation — 2 generators on 141 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,2,4)(6,11)(7,10)(8,9)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Polycyclic presentation:

2^2,7,17,113|a^2=b,c^a=c^6,d^a=d^16,e^a=e^112

Conjugacy classes & character table

13450 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");;
gap> coll := SingleCollector(F, [2, 2, 7, 17, 113]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13447 #72289340