Dic₆₉₅₀ #72236355

order	27800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₉₅₀

Splittings

Hall	C₂₅×C₁₃₉ ⋊ Q₈
Sylow	C₁₃₉ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₂₇₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₄₇₅
commutator, cosocle	C₂×C₂₅×C₁₃₉	C₂²
Frattini	C₂×C₅	C₂×D₆₉₅
Fitting	C₄×C₂₅×C₁₃₉	C₂
socle, prime radical	C₂×C₅×C₁₃₉	C₂×D₅
hypercenter	C₄	D₃₄₇₅
hypocenter	C₂₅×C₁₃₉	Q₈
LCS	G ▷ C₂×C₂₅×C₁₃₉ ▷ C₂₅×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 27800

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

Polycyclic presentation:

2^3,5,139,5|a^2=c,b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^138,f^a=f^4

Conjugacy classes & character table

6953 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₉₀₀
maximal normal subgroup of	Dic₁₃₉₀₀

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, 5, 139, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^138);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6950 #72236355