C₅₃×D₁₁₉ #27162197

order	12614
isoclinism	isoclinic class of D₁₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₁₁₉
unambiguous semidirect product	C₇⋊(D₁₇×C₅₃)
unambiguous semidirect product	C₁₇⋊(D₇×C₅₃)

Splittings

Fitting, socle, Hall	C₇×C₁₇×C₅₃ ⋊ C₂
derived, Hall	C₇×C₁₇ ⋊ C₂×C₅₃
Hall	C₁₇×C₅₃ ⋊ D₇
Hall	C₇×C₅₃ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₇×C₅₃
Sylow	C₇ ⋊ D₁₇×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₁₁₉
commutator, cosocle, hypocenter	C₇×C₁₇	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₅₃	C₂
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₅₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6307

A minimal faithful permutation representation — 2 generators on 77 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,20)(21,22)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,21,23,24,22,20)

Conjugacy classes & character table

3233 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₁₉, D₇×D₁₇×C₅₃, D₁₁₉⋊D₅₃, …
maximal normal subgroup of	D₅₃×D₁₁₉, D₇×D₁₇×C₅₃, D₁₁₉⋊D₅₃, …
maximal quotient of	C₅₃×Dic₁₁₉, C₂×C₅₃×D₁₁₉, C₃×C₅₃×D₁₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12614, 7);

C53×D119 #27162197