C₁₁₃×D₅₇ #27166264

order	12882
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁₃×D₅₇
unambiguous semidirect product	C₃⋊(D₁₉×C₁₁₃)
unambiguous semidirect product	C₁₉⋊(D₃×C₁₁₃)

Splittings

Fitting, socle, Hall	C₃×C₁₉×C₁₁₃ ⋊ C₂
derived, Hall	C₃×C₁₉ ⋊ C₂×C₁₁₃
Hall	C₁₉×C₁₁₃ ⋊ D₃
Hall	C₃×C₁₁₃ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₃×C₁₁₃
Sylow	C₃ ⋊ D₁₉×C₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁₃	D₅₇
commutator, cosocle, hypocenter	C₃×C₁₉	C₂×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₁₁₃	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₁₁₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6441

A minimal faithful permutation representation — 2 generators on 135 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,22)

Conjugacy classes & character table

3390 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₇×D₁₁₃, D₃×D₁₉×C₁₁₃, D₅₇⋊D₁₁₃, …
maximal normal subgroup of	D₅₇×D₁₁₃, D₃×D₁₉×C₁₁₃, D₅₇⋊D₁₁₃, …
maximal quotient of	C₁₁₃×Dic₅₇, C₂×C₁₁₃×D₅₇, C₁₁₃×D₁₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12882, 8);

C113×D57 #27166264