C₅₉⋊C₈ #90888

order	472
isoclinism	isoclinic class of D₅₉
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₅₉⋊C₈

Splittings

derived, Sylow

C₅₉ ⋊ C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₅₉
commutator, hypocenter	C₅₉	C₈
Fitting, cosocle	C₄×C₅₉	C₂
socle, prime radical	C₂×C₅₉	C₄
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₄

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 472

A minimal faithful permutation representation — 2 generators on 67 points:

(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)(60,61,62,64,63,65,66,67);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)

Polycyclic presentation:

2^3,59|a^2=b,b^2=c,d^a=d^58

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₅₉, #11745659, C₂×C₅₉⋊C₈, …
maximal normal subgroup of	C₈×D₅₉, #11745659, C₂×C₅₉⋊C₈, …
maximal quotient of	C₅₉⋊C₁₆, C₂×C₅₉⋊C₈, C₃×C₅₉⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(472, 1);

C59⋊C8 #90888