Dic₄₃₉₆ #72215855

order	17584
automorphism	Aut(G) order 32917248
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₉₆

Splittings

Hall	C₇×C₁₅₇ ⋊ Q₁₆
Sylow	C₁₅₇ ⋊ Dic₂₈
Sylow	C₇ ⋊ Dic₆₂₈

Subgroups

	subgroup	G/H
center	C₂	D₄₃₉₆
commutator, cosocle	C₄×C₇×C₁₅₇	C₂²
Frattini	C₄	C₂×D₁₀₉₉
Fitting	C₇×C₈×C₁₅₇	C₂
socle, prime radical	C₂×C₇×C₁₅₇	D₄
hypercenter	C₈	D₁₀₉₉
hypocenter	C₇×C₁₅₇	Q₁₆
LCS	G ▷ C₄×C₇×C₁₅₇ ▷ C₂×C₇×C₁₅₇ ▷ C₇×C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 180
transitive degree	μ_t(G) = 17584

A minimal faithful permutation representation — 2 generators on 180 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,180)(26,179)(27,178)(28,177)(29,176)(30,175)(31,174)(32,173)(33,172)(34,171)(35,170)(36,169)(37,168)(38,167)(39,166)(40,165)(41,164)(42,163)(43,162)(44,161)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2^4,7,157|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^156

Conjugacy classes & character table

4399 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₉₂, Dic₁₃₁₈₈
maximal normal subgroup of	Dic₈₇₉₂
maximal quotient of	Dic₁₃₁₈₈

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, 2, 7, 157]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^156);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4396 #72215855