Dic₇₈₀₀ #72272226

order	31200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₈₀₀

Splittings

Hall	C₃×C₁₃×C₂₅ ⋊ Q₃₂
Hall	C₁₃×C₂₅ ⋊ Dic₂₄
Hall	C₃×C₂₅ ⋊ Dic₁₀₄
Hall	C₃×C₁₃ ⋊ Dic₂₀₀
Sylow	C₂₅ ⋊ Dic₃₁₂
Sylow	C₁₃ ⋊ Dic₆₀₀
Sylow	C₃ ⋊ Dic₂₆₀₀

Subgroups

	subgroup	G/H
center	C₂	D₇₈₀₀
commutator, cosocle	C₃×C₈×C₁₃×C₂₅	C₂²
Frattini	C₅×C₈	C₂×D₁₉₅
Fitting	C₃×C₁₃×C₁₆×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₁₃	D₄₀
hypercenter	C₁₆	D₉₇₅
hypocenter	C₃×C₁₃×C₂₅	Q₃₂
LCS	G ▷ C₃×C₈×C₁₃×C₂₅ ▷ C₃×C₄×C₁₃×C₂₅ ▷ C₂×C₃×C₁₃×C₂₅ ▷ C₃×C₁₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 31200

A minimal faithful permutation representation — 2 generators on 73 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^5,3,5,13,5|a^2=e,b^2=cd,c^2=de,d^2=e,g^5=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^4i^4,h^a=h^12,i^a=i^4

Conjugacy classes & character table

7803 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₆₀₀
maximal normal subgroup of	Dic₁₅₆₀₀

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 13, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 7, F.9);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);