D₁₂₁₂₁ #37564598

order	24242
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₁₂₁
unambiguous semidirect product	C₁₇⋊D₇₁₃
unambiguous semidirect product	C₂₃⋊D₅₂₇
unambiguous semidirect product	C₃₁⋊D₃₉₁

Splittings

Fitting, socle, derived, Hall	C₁₇×C₂₃×C₃₁ ⋊ C₂
Hall	C₂₃×C₃₁ ⋊ D₁₇
Hall	C₁₇×C₃₁ ⋊ D₂₃
Hall	C₁₇×C₂₃ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₃₉₁
Sylow	C₂₃ ⋊ D₅₂₇
Sylow	C₁₇ ⋊ D₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₇×C₂₃×C₃₁	C₂
LCS	G ▷ C₁₇×C₂₃×C₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 12121

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);(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(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:

2,17,23,31|b^a=b^16,c^a=c^22,d^a=d^30

Conjugacy classes & character table

6062 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24242, 8);

D12121 #37564598