Dic₃₁₆₄ #72213477

order	12656
automorphism	Aut(G) order 17009664
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₆₄

Splittings

Hall	C₇×C₁₁₃ ⋊ Q₁₆
Sylow	C₁₁₃ ⋊ Dic₂₈
Sylow	C₇ ⋊ Dic₄₅₂

Subgroups

	subgroup	G/H
center	C₂	D₃₁₆₄
commutator, cosocle	C₄×C₇×C₁₁₃	C₂²
Frattini	C₄	C₂×D₇₉₁
Fitting	C₇×C₈×C₁₁₃	C₂
socle, prime radical	C₂×C₇×C₁₁₃	D₄
hypercenter	C₈	D₇₉₁
hypocenter	C₇×C₁₁₃	Q₁₆
LCS	G ▷ C₄×C₇×C₁₁₃ ▷ C₂×C₇×C₁₁₃ ▷ C₇×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 12656

A minimal faithful permutation representation — 2 generators on 136 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2^4,7,113|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^112

Conjugacy classes & character table

3167 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₃₂₈, Dic₉₄₉₂, Dic₁₅₈₂₀
maximal normal subgroup of	Dic₆₃₂₈
maximal quotient of	Dic₉₄₉₂, Dic₁₅₈₂₀

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, 113]);;
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.6^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3164 #72213477