C₃×C₇×D₃₂₃ #27214483

order	13566
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₇×D₃₂₃
unambiguous semidirect product	C₁₇⋊(C₃×C₇×D₁₉)

Splittings

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	D₃₂₃
commutator, cosocle, hypocenter	C₁₇×C₁₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₇×C₁₉	C₂
LCS	G ▷ C₁₇×C₁₉
UCS	C₁ ◁ C₃×C₇

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 6783

A minimal faithful permutation representation — 2 generators on 46 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(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);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,23,25,27,29,31,33,35,36,34,32,30,28,26,24,22)

Conjugacy classes & character table

3423 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₇×D₁₇×D₁₉, C₃×D₇×D₃₂₃, C₃×D₁₁₉⋊D₁₉, …
maximal normal subgroup of	C₃×C₇×D₁₇×D₁₉, C₃×D₇×D₃₂₃, C₃×D₁₁₉⋊D₁₉, …
maximal quotient of	C₃×C₇×Dic₃₂₃, C₂×C₃×C₇×D₃₂₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13566, 8);

Fitting, socle, Hall	C₃×C₇×C₁₇×C₁₉ ⋊ C₂
derived, Hall	C₁₇×C₁₉ ⋊ C₂×C₃×C₇
Hall	C₇×C₁₇×C₁₉ ⋊ C₂×C₃
Hall	C₃×C₁₇×C₁₉ ⋊ C₂×C₇
Hall	C₃×C₇×C₁₉ ⋊ D₁₇
Hall	C₃×C₇×C₁₇ ⋊ D₁₉
Hall	C₇×C₁₉ ⋊ C₃×D₁₇
Hall	C₇×C₁₇ ⋊ C₃×D₁₉
Hall	C₃×C₁₉ ⋊ C₇×D₁₇
Hall	C₃×C₁₇ ⋊ C₇×D₁₉
Sylow	C₁₉ ⋊ C₃×C₇×D₁₇
Sylow	C₁₇ ⋊ C₃×C₇×D₁₉

C3×C7×D323 #27214483