Dic₈₀₇₅ #44568019

order	32300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₇₅
unambiguous semidirect product	C₁₉⋊Dic₄₂₅

Splittings

derived, Hall	C₁₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₁₉×C₂₅ ⋊ Dic₁₇
Hall	C₁₇×C₂₅ ⋊ Dic₁₉
Hall	C₁₇×C₁₉ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₃₂₃
Sylow	C₁₉ ⋊ Dic₄₂₅
Sylow	C₁₇ ⋊ Dic₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₀₇₅
commutator, hypocenter	C₁₇×C₁₉×C₂₅	C₄
Frattini	C₂×C₅	D₁₆₁₅
Fitting, cosocle	C₂×C₁₇×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₁₇×C₁₉	D₅
LCS	G ▷ C₁₇×C₁₉×C₂₅
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 32300

A minimal faithful permutation representation — 2 generators on 65 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);(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)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,5,17,19,2,5|a^2=e,b^5=f^4,b^a=b^4f,c^a=c^16,d^a=d^18,f^a=f^4

Conjugacy classes & character table

8078 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₁₅₀
maximal normal subgroup of	Dic₁₆₁₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32300, 8);

Dic8075 #44568019