Dic₁₄₄₇₈ #72291200

order	57912
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₄₇₈

Splittings

Hall	C₃×C₁₉×C₁₂₇ ⋊ Q₈
Hall	C₁₉×C₁₂₇ ⋊ Dic₆
Hall	C₃×C₁₂₇ ⋊ Dic₃₈
Sylow	C₁₂₇ ⋊ Dic₁₁₄
Hall	C₃×C₁₉ ⋊ Dic₂₅₄
Sylow	C₁₉ ⋊ Dic₇₆₂
Sylow	C₃ ⋊ Dic₄₈₂₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₂₃₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₁₂₇	C₂²
Fitting	C₃×C₄×C₁₉×C₁₂₇	C₂
hypercenter	C₄	D₇₂₃₉
hypocenter	C₃×C₁₉×C₁₂₇	Q₈
LCS	G ▷ C₂×C₃×C₁₉×C₁₂₇ ▷ C₃×C₁₉×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 57912

A minimal faithful permutation representation — 2 generators on 157 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,5,3,7)(2,6,4,8)(10,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)

Polycyclic presentation:

2^3,3,19,127|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^18,f^a=f^126

Conjugacy classes & character table

14481 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, 3, 19, 127]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14478 #72291200