Dic₅₁₁₇ #35205950

order	20468
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₁₇
unambiguous semidirect product	C₇⋊Dic₇₃₁
unambiguous semidirect product	C₄₃⋊Dic₁₁₉

Splittings

derived, Hall	C₇×C₁₇×C₄₃ ⋊ C₄
Hall	C₁₇×C₄₃ ⋊ Dic₇
Hall	C₇×C₄₃ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₃₀₁
Sylow	C₇ ⋊ Dic₇₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₁₇
commutator, hypocenter	C₇×C₁₇×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₄₃	C₂
LCS	G ▷ C₇×C₁₇×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 20468

A minimal faithful permutation representation — 2 generators on 71 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,7,17,43,2|a^2=e,b^a=b^6,c^a=c^16,d^a=d^42

Conjugacy classes & character table

5120 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₂₃₄, Dic₁₅₃₅₁
maximal normal subgroup of	Dic₁₀₂₃₄
maximal quotient of	Dic₁₅₃₅₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20468, 9);

Dic5117 #35205950