Dic₇₄₀₀ #72236616

order	29600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₀₀

Splittings

Hall	C₂₅×C₃₇ ⋊ Q₃₂
Sylow	C₃₇ ⋊ Dic₂₀₀
Sylow	C₂₅ ⋊ Dic₂₉₆

Subgroups

	subgroup	G/H
center	C₂	D₇₄₀₀
commutator, cosocle	C₈×C₂₅×C₃₇	C₂²
Frattini	C₅×C₈	C₂×D₁₈₅
Fitting	C₁₆×C₂₅×C₃₇	C₂
socle, prime radical	C₂×C₅×C₃₇	D₄₀
hypercenter	C₁₆	D₉₂₅
hypocenter	C₂₅×C₃₇	Q₃₂
LCS	G ▷ C₈×C₂₅×C₃₇ ▷ C₄×C₂₅×C₃₇ ▷ C₂×C₂₅×C₃₇ ▷ C₂₅×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 29600

A minimal faithful permutation representation — 2 generators on 94 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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94);(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)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^5,5,37,5|a^2=e,b^2=cd,c^2=de,d^2=e,f^5=h,b^a=bc,c^a=cd,d^a=de,f^a=f^4h^4,g^a=g^36,h^a=h^4

Conjugacy classes & character table

7403 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₈₀₀
maximal normal subgroup of	Dic₁₄₈₀₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 37, 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, 6, F.8);;
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.6^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7400 #72236616