Dic₄₄₂₅ #31799147

order	17700
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₄₂₅
unambiguous semidirect product	C₃⋊Dic₁₄₇₅
unambiguous semidirect product	C₅₉⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₅₉ ⋊ C₄
Hall	C₂₅×C₅₉ ⋊ Dic₃
Hall	C₃×C₅₉ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₁₇₇
Sylow	C₃ ⋊ Dic₁₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₄₂₅
commutator, hypocenter	C₃×C₂₅×C₅₉	C₄
Frattini	C₂×C₅	D₈₈₅
Fitting, cosocle	C₂×C₃×C₂₅×C₅₉	C₂
socle, prime radical	C₂×C₃×C₅×C₅₉	D₅
LCS	G ▷ C₃×C₂₅×C₅₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 17700

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

Polycyclic presentation:

2,3,5,59,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^58,f^a=f^4

Conjugacy classes & character table

4428 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₅₀, Dic₁₃₂₇₅
maximal normal subgroup of	Dic₈₈₅₀
maximal quotient of	Dic₁₃₂₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17700, 8);

Dic4425 #31799147