Dic₇₃₉₈ #72236612

order	29592
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₃₉₈

Splittings

Hall	C₂₇×C₁₃₇ ⋊ Q₈
Sylow	C₁₃₇ ⋊ Dic₅₄
Sylow	C₂₇ ⋊ Dic₂₇₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₆₉₉
commutator, cosocle	C₂×C₂₇×C₁₃₇	C₂²
Frattini	C₂×C₉	C₂×D₄₁₁
Fitting	C₄×C₂₇×C₁₃₇	C₂
socle, prime radical	C₂×C₃×C₁₃₇	C₂×D₉
hypercenter	C₄	D₃₆₉₉
hypocenter	C₂₇×C₁₃₇	Q₈
LCS	G ▷ C₂×C₂₇×C₁₃₇ ▷ C₂₇×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 29592

A minimal faithful permutation representation — 2 generators on 172 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)

Polycyclic presentation:

2^3,3,137,3^2|a^2=c,b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^136,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

7401 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₇₉₆
maximal normal subgroup of	Dic₁₄₇₉₆

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 137, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^136);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7398 #72236612