Dic₆₄₁₇ #38787262

order	25668
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₄₁₇
unambiguous semidirect product	C₉⋊Dic₇₁₃
unambiguous semidirect product	C₂₃⋊Dic₂₇₉
unambiguous semidirect product	C₃₁⋊Dic₂₀₇

Splittings

derived, Hall	C₉×C₂₃×C₃₁ ⋊ C₄
Hall	C₂₃×C₃₁ ⋊ Dic₉
Hall	C₉×C₃₁ ⋊ Dic₂₃
Hall	C₉×C₂₃ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₀₇
Sylow	C₂₃ ⋊ Dic₂₇₉
Sylow	C₉ ⋊ Dic₇₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₄₁₇
commutator, hypocenter	C₉×C₂₃×C₃₁	C₄
Frattini	C₂×C₃	D₂₁₃₉
Fitting, cosocle	C₂×C₉×C₂₃×C₃₁	C₂
socle, prime radical	C₂×C₃×C₂₃×C₃₁	D₃
LCS	G ▷ C₉×C₂₃×C₃₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 25668

A minimal faithful permutation representation — 2 generators on 67 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,3,23,31,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^22,d^a=d^30,f^a=f^2

Conjugacy classes & character table

6420 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₈₃₄
maximal normal subgroup of	Dic₁₂₈₃₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25668, 9);

Dic6417 #38787262