Dic₇₃₃₆ #72236554

order	29344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₃₃₆

Splittings

Hall	C₇×C₁₃₁ ⋊ Q₃₂
Sylow	C₁₃₁ ⋊ Dic₅₆
Sylow	C₇ ⋊ Dic₁₀₄₈

Subgroups

	subgroup	G/H
center	C₂	D₇₃₃₆
commutator, cosocle	C₇×C₈×C₁₃₁	C₂²
Frattini	C₈	C₂×D₉₁₇
Fitting	C₇×C₁₆×C₁₃₁	C₂
socle, prime radical	C₂×C₇×C₁₃₁	D₈
hypercenter	C₁₆	D₉₁₇
hypocenter	C₇×C₁₃₁	Q₃₂
LCS	G ▷ C₇×C₈×C₁₃₁ ▷ C₄×C₇×C₁₃₁ ▷ C₂×C₇×C₁₃₁ ▷ C₇×C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 170
transitive degree	μ_t(G) = 29344

A minimal faithful permutation representation — 2 generators on 170 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,39)(35,38)(36,37)(41,170)(42,169)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,150)(62,149)(63,148)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)

Polycyclic presentation:

2^5,7,131|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^130

Conjugacy classes & character table

7339 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₆₇₂
maximal normal subgroup of	Dic₁₄₆₇₂

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, 2, 2, 7, 131]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^130);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7336 #72236554