Dic₅₁₅₀ #72216786

order	20600
automorphism	Aut(G) order 42024000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₁₅₀

Splittings

Hall	C₂₅×C₁₀₃ ⋊ Q₈
Sylow	C₁₀₃ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₂₀₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₅₇₅
commutator, cosocle	C₂×C₂₅×C₁₀₃	C₂²
Frattini	C₂×C₅	C₂×D₅₁₅
Fitting	C₄×C₂₅×C₁₀₃	C₂
socle, prime radical	C₂×C₅×C₁₀₃	C₂×D₅
hypercenter	C₄	D₂₅₇₅
hypocenter	C₂₅×C₁₀₃	Q₈
LCS	G ▷ C₂×C₂₅×C₁₀₃ ▷ C₂₅×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 20600

A minimal faithful permutation representation — 2 generators on 136 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);(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)(35,136)(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:

2^3,5,103,5|a^2=c,b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^102,f^a=f^4

Conjugacy classes & character table

5153 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₀₀, Dic₁₅₄₅₀
maximal normal subgroup of	Dic₁₀₃₀₀
maximal quotient of	Dic₁₅₄₅₀

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, 103, 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.5^102);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5150 #72216786