Dic₇₀₁₄ #72236384

order	28056
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₀₁₄

Splittings

Hall	C₃×C₇×C₁₆₇ ⋊ Q₈
Hall	C₇×C₁₆₇ ⋊ Dic₆
Hall	C₃×C₁₆₇ ⋊ Dic₁₄
Sylow	C₁₆₇ ⋊ Dic₄₂
Hall	C₃×C₇ ⋊ Dic₃₃₄
Sylow	C₇ ⋊ Dic₁₀₀₂
Sylow	C₃ ⋊ Dic₂₃₃₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₅₀₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₇×C₁₆₇	C₂²
Fitting	C₃×C₄×C₇×C₁₆₇	C₂
hypercenter	C₄	D₃₅₀₇
hypocenter	C₃×C₇×C₁₆₇	Q₈
LCS	G ▷ C₂×C₃×C₇×C₁₆₇ ▷ C₃×C₇×C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 185
transitive degree	μ_t(G) = 28056

A minimal faithful permutation representation — 2 generators on 185 points:

(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(1,5,3,7)(2,6,4,8)(10,11)(13,18)(14,17)(15,16)(20,185)(21,184)(22,183)(23,182)(24,181)(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:

2^3,3,7,167|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^166

Conjugacy classes & character table

7017 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₀₂₈
maximal normal subgroup of	Dic₁₄₀₂₈

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, 7, 167]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7014 #72236384