C₄₁×C₄₃⋊C₇ #27134643

order	12341
isoclinism	isoclinic class of C₄₃⋊C₇
automorphism	?
homology	Abelianization (H₁) = C₇×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₁×C₄₃⋊C₇
unambiguous semidirect product	C₄₃⋊(C₇×C₄₁)
unambiguous semidirect product	(C₄₁×C₄₃)⋊C₇

Splittings

Fitting, socle, Hall	C₄₁×C₄₃ ⋊ C₇
derived, Sylow	C₄₃ ⋊ C₇×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	C₄₃⋊C₇
commutator, cosocle, hypocenter	C₄₃	C₇×C₄₁
Frattini	C₁	G
Fitting, socle	C₄₁×C₄₃	C₇
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₄₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 1763

A minimal faithful permutation representation — 2 generators on 84 points:

(2,3,5,9,14,22,35)(4,7,11,18,29,40,6)(8,13,20,33,10,16,25)(12,17,27,38,21,32,43)(15,24,37,34,23,19,31)(26,36,28,39,30,41,42)(44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(1,2,4,8,5,10,17,28,11,19,32,22,20,34,24,37,14,23,36,33,25,35,40,43,38,31,42,18,30,39,41,27,29,16,26,9,15,13,21,7,12,3,6)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₄₁×C₄₃⋊C₇, C₄₁×D₄₃⋊C₇, D₄₁×C₄₃⋊C₇, …
maximal normal subgroup of	C₂×C₄₁×C₄₃⋊C₇, C₄₁×D₄₃⋊C₇, D₄₁×C₄₃⋊C₇, …
maximal quotient of	C₂×C₄₁×C₄₃⋊C₇, C₃×C₄₁×C₄₃⋊C₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12341, 1);

C41×C43⋊C7 #27134643