D₇₇₈₄ #72212909

order	15568
automorphism	Aut(G) order 25780608
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₇₈₄

Splittings

Fitting	C₇×C₈×C₁₃₉ ⋊ C₂
Hall	C₇×C₁₃₉ ⋊ D₈
Sylow	C₁₃₉ ⋊ D₅₆
Sylow	C₇ ⋊ D₁₁₁₂

Subgroups

	subgroup	G/H
center	C₂	D₃₈₉₂
commutator, cosocle	C₄×C₇×C₁₃₉	C₂²
Frattini	C₄	C₂×D₉₇₃
Fitting	C₇×C₈×C₁₃₉	C₂
socle	C₂×C₇×C₁₃₉	D₄
hypercenter	C₈	D₉₇₃
hypocenter	C₇×C₁₃₉	D₈
LCS	G ▷ C₄×C₇×C₁₃₉ ▷ C₂×C₇×C₁₃₉ ▷ C₇×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 154
transitive degree	μ_t(G) = 7784

A minimal faithful permutation representation — 2 generators on 154 points:

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

Polycyclic presentation:

2^4,7,139|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^138

Conjugacy classes & character table

3895 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₅₅₆₈, D₂₃₃₅₂
maximal normal subgroup of	D₁₅₅₆₈
maximal quotient of	Dic₇₇₈₄, D₁₅₅₆₈, D₂₃₃₅₂

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, 139]);;
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.6^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7784 #72212909