Dic₅₄₇₅ #36346381

order	21900
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₄₇₅
unambiguous semidirect product	C₃⋊Dic₁₈₂₅

Splittings

derived, Hall	C₃×C₂₅×C₇₃ ⋊ C₄
Hall	C₂₅×C₇₃ ⋊ Dic₃
Hall	C₃×C₇₃ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₂₁₉
Sylow	C₃ ⋊ Dic₁₈₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₄₇₅
commutator, hypocenter	C₃×C₂₅×C₇₃	C₄
Frattini	C₂×C₅	D₁₀₉₅
Fitting, cosocle	C₂×C₃×C₂₅×C₇₃	C₂
socle, prime radical	C₂×C₃×C₅×C₇₃	D₅
LCS	G ▷ C₃×C₂₅×C₇₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 21900

A minimal faithful permutation representation — 2 generators on 105 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);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2,3,5,73,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^72,f^a=f^4

Conjugacy classes & character table

5478 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₉₅₀
maximal normal subgroup of	Dic₁₀₉₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21900, 9);

Dic5475 #36346381