Dic₄₂₅₇ #30684480

order	17028
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₂₅₇
unambiguous semidirect product	C₉⋊Dic₄₇₃
unambiguous semidirect product	C₁₁⋊Dic₃₈₇
unambiguous semidirect product	C₄₃⋊Dic₉₉

Splittings

derived, Hall	C₉×C₁₁×C₄₃ ⋊ C₄
Hall	C₁₁×C₄₃ ⋊ Dic₉
Hall	C₉×C₄₃ ⋊ Dic₁₁
Hall	C₉×C₁₁ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₉₉
Sylow	C₁₁ ⋊ Dic₃₈₇
Sylow	C₉ ⋊ Dic₄₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₂₅₇
commutator, hypocenter	C₉×C₁₁×C₄₃	C₄
Frattini	C₂×C₃	D₁₄₁₉
Fitting, cosocle	C₂×C₉×C₁₁×C₄₃	C₂
socle, prime radical	C₂×C₃×C₁₁×C₄₃	D₃
LCS	G ▷ C₉×C₁₁×C₄₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 17028

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)(15,24)(16,23)(17,22)(18,21)(19,20)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,3,11,43,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^42,f^a=f^2

Conjugacy classes & character table

4260 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₅₁₄, Dic₁₂₇₇₁
maximal normal subgroup of	Dic₈₅₁₄
maximal quotient of	Dic₁₂₇₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17028, 9);

Dic4257 #30684480