Dic₆₂₂₅ #38735199

order	24900
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₂₂₅
unambiguous semidirect product	C₃⋊Dic₂₀₇₅
unambiguous semidirect product	C₈₃⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₈₃ ⋊ C₄
Hall	C₂₅×C₈₃ ⋊ Dic₃
Hall	C₃×C₈₃ ⋊ Dic₂₅
Sylow	C₈₃ ⋊ Dic₇₅
Hall	C₃×C₂₅ ⋊ Dic₈₃
Sylow	C₂₅ ⋊ Dic₂₄₉
Sylow	C₃ ⋊ Dic₂₀₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₂₂₅
commutator, hypocenter	C₃×C₂₅×C₈₃	C₄
Frattini	C₂×C₅	D₁₂₄₅
Fitting, cosocle	C₂×C₃×C₂₅×C₈₃	C₂
socle, prime radical	C₂×C₃×C₅×C₈₃	D₅
LCS	G ▷ C₃×C₂₅×C₈₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 24900

A minimal faithful permutation representation — 2 generators on 115 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(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)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2,3,5,83,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^82,f^a=f^4

Conjugacy classes & character table

6228 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₄₅₀
maximal normal subgroup of	Dic₁₂₄₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24900, 8);

Dic6225 #38735199