C₇×D₃₂₃ #17985285

order	4522
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₃₂₃
unambiguous semidirect product	C₁₇⋊(C₇×D₁₉)
unambiguous semidirect product	C₁₉⋊(C₇×D₁₇)

Splittings

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

Subgroups

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

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 2261

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

1141 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₇×D₃₂₃, C₇×D₉₆₉, C₇×C₁₇⋊(D₁₉⋊C₃), …
maximal normal subgroup of	C₃×C₇×D₃₂₃, C₇×C₁₇⋊(D₁₉⋊C₃), ΓL(1,𝔽₈)×D₃₂₃, …
maximal quotient of	C₃×C₇×D₃₂₃, C₇×D₉₆₉, C₅×C₇×D₃₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4522, 4);

C7×D323 #17985285