Dic₆₃₂₅ #38760673

order	25300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₃₂₅
unambiguous semidirect product	C₁₁⋊Dic₅₇₅
unambiguous semidirect product	C₂₃⋊Dic₂₇₅

Splittings

derived, Hall	C₁₁×C₂₃×C₂₅ ⋊ C₄
Hall	C₂₃×C₂₅ ⋊ Dic₁₁
Hall	C₁₁×C₂₅ ⋊ Dic₂₃
Hall	C₁₁×C₂₃ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₂₅₃
Sylow	C₂₃ ⋊ Dic₂₇₅
Sylow	C₁₁ ⋊ Dic₅₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₃₂₅
commutator, hypocenter	C₁₁×C₂₃×C₂₅	C₄
Frattini	C₂×C₅	D₁₂₆₅
Fitting, cosocle	C₂×C₁₁×C₂₃×C₂₅	C₂
socle, prime radical	C₂×C₅×C₁₁×C₂₃	D₅
LCS	G ▷ C₁₁×C₂₃×C₂₅
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 25300

A minimal faithful permutation representation — 2 generators on 63 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);(1,3,2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,40)(32,39)(33,38)(34,37)(35,36)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,5,11,23,2,5|a^2=e,b^5=f^4,b^a=b^4f,c^a=c^10,d^a=d^22,f^a=f^4

Conjugacy classes & character table

6328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₆₅₀
maximal normal subgroup of	Dic₁₂₆₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25300, 10);

Dic6325 #38760673