C₃⋊C₈×C₄₉ #11924088

order	1176
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₈×C₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃⋊C₈×C₄₉
unambiguous semidirect product	C₃⋊(C₈×C₄₉)

Splittings

derived, Sylow	C₃ ⋊ C₈×C₄₉
Hall	C₃×C₄₉ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₄₉	D₃
commutator, hypocenter	C₃	C₈×C₄₉
Frattini	C₄×C₇	D₃×C₇
Fitting	C₃×C₄×C₄₉	C₂
socle, prime radical	C₂×C₃×C₇	C₄×C₇
cosocle	C₃×C₄×C₇	C₂×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₄₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 1176

A minimal faithful permutation representation — 2 generators on 60 points:

(2,3)(4,5,6,8,7,9,10,11)(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);(1,2,3)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 3);

C3⋊C8×C49 #11924088