C₃×D₇×C₄₃⋊C₇ #27162385

order	12642
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₇×C₄₃⋊C₇
unambiguous semidirect product	(C₃×D₇×C₄₃)⋊C₇

Splittings

Fitting, socle	C₃×C₇×C₄₃ ⋊ C₂×C₇
derived	C₇×C₄₃ ⋊ C₂×C₃×C₇
Hall	C₃×C₇×C₄₃⋊C₇ ⋊ C₂
Hall	C₇×C₄₃⋊C₇ ⋊ C₂×C₃
Hall	C₃×C₄₃ ⋊ C₇×D₇
Sylow	C₄₃ ⋊ C₃×C₇×D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₇×C₄₃⋊C₇
commutator, cosocle, hypocenter	C₇×C₄₃	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₄₃	C₂×C₇
LCS	G ▷ C₇×C₄₃
UCS	C₁ ◁ C₃

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 903

A minimal faithful permutation representation — 2 generators on 53 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)(45,46)(47,48)(49,50)(51,52,53);(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)(44,45,47,49,50,48,46)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₇×D₄₃⋊C₇, D₇×C₃⋊(D₄₃⋊C₇), D₃×D₇×C₄₃⋊C₇, …
maximal normal subgroup of	C₃×D₇×D₄₃⋊C₇, D₇×C₃⋊(D₄₃⋊C₇), D₃×D₇×C₄₃⋊C₇, …
maximal quotient of	C₃×Dic₇×C₄₃⋊C₇, C₂×C₃×D₇×C₄₃⋊C₇, C₉×D₇×C₄₃⋊C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 50);

C3×D7×C43⋊C7 #27162385

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×D₇×C₄₃⋊C₇ #27162385