C₃×C₇×D₄₃⋊C₇ #27162383

order	12642
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₃×C₇×D₄₃⋊C₇
unambiguous semidirect product	(C₃×D₄₃)⋊C₇²
unambiguous semidirect product	(C₃×C₇×D₄₃)⋊C₇

Splittings

Fitting, socle	C₃×C₇×C₄₃ ⋊ C₂×C₇
derived, Sylow	C₄₃ ⋊ C₂×C₃×C₇²
Hall	C₃×C₇×C₄₃⋊C₇ ⋊ C₂
Hall	C₇×C₄₃⋊C₇ ⋊ C₂×C₃
Hall	C₃×D₄₃ ⋊ C₇²
Hall	C₃×C₄₃ ⋊ C₂×C₇²
Hall	D₄₃ ⋊ C₃×C₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	D₄₃⋊C₇
commutator, cosocle, hypocenter	C₄₃	C₂×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₄₃	C₂×C₇
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₃×C₇

80 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×D₇×D₄₃⋊C₇, D₃×C₇×D₄₃⋊C₇, D₂₁×D₄₃⋊C₇, …
maximal normal subgroup of	C₃×D₇×D₄₃⋊C₇, D₃×C₇×D₄₃⋊C₇, D₂₁×D₄₃⋊C₇, …
maximal quotient of	C₃×C₇×Dic₄₃⋊C₇, C₂×C₃×C₇×D₄₃⋊C₇, C₇×C₉×D₄₃⋊C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 48);

C3×C7×D43⋊C7 #27162383