Dic₄₅₇₅ #32912895

order	18300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₇₅
unambiguous semidirect product	C₃⋊Dic₁₅₂₅

Splittings

derived, Hall	C₃×C₂₅×C₆₁ ⋊ C₄
Hall	C₂₅×C₆₁ ⋊ Dic₃
Hall	C₃×C₆₁ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₁₈₃
Sylow	C₃ ⋊ Dic₁₅₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₅₇₅
commutator, hypocenter	C₃×C₂₅×C₆₁	C₄
Frattini	C₂×C₅	D₉₁₅
Fitting, cosocle	C₂×C₃×C₂₅×C₆₁	C₂
socle, prime radical	C₂×C₃×C₅×C₆₁	D₅
LCS	G ▷ C₃×C₂₅×C₆₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 18300

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

Polycyclic presentation:

2,3,5,61,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^60,f^a=f^4

Conjugacy classes & character table

4578 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₁₅₀, Dic₁₃₇₂₅
maximal normal subgroup of	Dic₉₁₅₀
maximal quotient of	Dic₁₃₇₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18300, 18);

Dic4575 #32912895