D₃₀₉₆₈ #72297846

order	61936
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₉₆₈

Splittings

Fitting	C₈×C₄₉×C₇₉ ⋊ C₂
Hall	C₄₉×C₇₉ ⋊ D₈
Sylow	C₇₉ ⋊ D₃₉₂
Sylow	C₄₉ ⋊ D₆₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₈₄
commutator, cosocle	C₄×C₄₉×C₇₉	C₂²
Frattini	C₄×C₇	C₂×D₅₅₃
Fitting	C₈×C₄₉×C₇₉	C₂
socle	C₂×C₇×C₇₉	D₂₈
hypercenter	C₈	D₃₈₇₁
hypocenter	C₄₉×C₇₉	D₈
LCS	G ▷ C₄×C₄₉×C₇₉ ▷ C₂×C₄₉×C₇₉ ▷ C₄₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 30968

A minimal faithful permutation representation — 2 generators on 136 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);(2,8)(3,7)(4,6)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^4,7,79,7|b^2=cd,c^2=d,e^7=g,b^a=bc,c^a=cd,e^a=e^6g^6,f^a=f^78,g^a=g^6

Conjugacy classes & character table

15487 conjugacy classes. The character table is too large to display.

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, 7, 79, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
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*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30968 #72297846