C₁₇×#10624391 #27134654

order	12342
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₇×#10624391

Splittings

Fitting, socle, Hall	C₁₁²×C₁₇ ⋊ D₃
derived, Hall	C₁₁²⋊C₃ ⋊ C₂×C₁₇
Hall	C₁₇×C₁₁²⋊C₃ ⋊ C₂
Sylow	C₁₁² ⋊ D₃×C₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₇	#10624391
commutator, cosocle, hypocenter	C₁₁²⋊C₃	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₁₁²×C₁₇	D₃
LCS	G ▷ C₁₁²⋊C₃
UCS	C₁ ◁ C₁₇

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 561

A minimal faithful permutation representation — 2 generators on 50 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₇×#10624391⋊C₂, #10624391⋊D₁₇, D₁₇×#10624391, …
maximal normal subgroup of	C₁₇×#10624391⋊C₂, #10624391⋊D₁₇, D₁₇×#10624391, …
maximal quotient of	C₁₇×#13089712, C₂×C₁₇×#10624391, C₁₇×#14546095, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12342, 10);

C17×#10624391 #27134654