D₃₀₇₃₆ #72296071

order	61472
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₇₃₆

Splittings

Fitting	C₁₆×C₁₇×C₁₁₃ ⋊ C₂
Hall	C₁₇×C₁₁₃ ⋊ D₁₆
Sylow	C₁₁₃ ⋊ D₂₇₂
Sylow	C₁₇ ⋊ D₁₈₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₃₆₈
commutator, cosocle	C₈×C₁₇×C₁₁₃	C₂²
Frattini	C₈	C₂×D₁₉₂₁
Fitting	C₁₆×C₁₇×C₁₁₃	C₂
socle	C₂×C₁₇×C₁₁₃	D₈
hypercenter	C₁₆	D₁₉₂₁
hypocenter	C₁₇×C₁₁₃	D₁₆
LCS	G ▷ C₈×C₁₇×C₁₁₃ ▷ C₄×C₁₇×C₁₁₃ ▷ C₂×C₁₇×C₁₁₃ ▷ C₁₇×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 146
transitive degree	μ_t(G) = 30736

A minimal faithful permutation representation — 2 generators on 146 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)

Polycyclic presentation:

2^5,17,113|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^16,g^a=g^112

Conjugacy classes & character table

15371 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, 2, 17, 113]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30736 #72296071