Dic₆₂₉₃ #38737744

order	25172
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₂₉₃
unambiguous semidirect product	C₇⋊Dic₈₉₉
unambiguous semidirect product	C₃₁⋊Dic₂₀₃

Splittings

derived, Hall	C₇×C₂₉×C₃₁ ⋊ C₄
Hall	C₂₉×C₃₁ ⋊ Dic₇
Hall	C₇×C₃₁ ⋊ Dic₂₉
Hall	C₇×C₂₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₀₃
Sylow	C₂₉ ⋊ Dic₂₁₇
Sylow	C₇ ⋊ Dic₈₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₂₉₃
commutator, hypocenter	C₇×C₂₉×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₉×C₃₁	C₂
LCS	G ▷ C₇×C₂₉×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 25172

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)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,7,29,31,2|a^2=e,b^a=b^6,c^a=c^28,d^a=d^30

Conjugacy classes & character table

6296 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₅₈₆
maximal normal subgroup of	Dic₁₂₅₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25172, 9);

Dic6293 #38737744