Dic₃₂₂₅ #27166343

order	12900
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₂₂₅
unambiguous semidirect product	C₃⋊Dic₁₀₇₅
unambiguous semidirect product	C₄₃⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₄₃ ⋊ C₄
Hall	C₂₅×C₄₃ ⋊ Dic₃
Hall	C₃×C₄₃ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₁₂₉
Sylow	C₃ ⋊ Dic₁₀₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₂₅
commutator, hypocenter	C₃×C₂₅×C₄₃	C₄
Frattini	C₂×C₅	D₆₄₅
Fitting, cosocle	C₂×C₃×C₂₅×C₄₃	C₂
socle, prime radical	C₂×C₃×C₅×C₄₃	D₅
LCS	G ▷ C₃×C₂₅×C₄₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 12900

A minimal faithful permutation representation — 2 generators on 75 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);(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)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,3,5,43,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^42,f^a=f^4

Conjugacy classes & character table

3228 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₄₅₀, Dic₁₆₁₂₅
maximal normal subgroup of	Dic₆₄₅₀
maximal quotient of	Dic₁₆₁₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 12);

Dic3225 #27166343