C₁₁×C₁₃²⋊C₇ #27189013

order	13013
automorphism	?
homology	Abelianization (H₁) = C₇×C₁₁ · Schur multiplier (H₂) = C₁₃

Aliases

direct product	C₁₁×C₁₃²⋊C₇
unambiguous semidirect product	C₁₃²⋊(C₇×C₁₁)
unambiguous semidirect product	(C₁₁×C₁₃²)⋊C₇

Splittings

Fitting, socle, Hall	C₁₁×C₁₃² ⋊ C₇
derived, Sylow	C₁₃² ⋊ C₇×C₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	C₁₃²⋊C₇
commutator, cosocle, hypocenter	C₁₃²	C₇×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₃²	C₇
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₁₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 1001

A minimal faithful permutation representation — 2 generators on 102 points:

(1,2,4,9,19,37,54)(3,7,16,33,38,57,75)(5,12,20,39,58,76,83)(6,14,28,48,67,82,89)(8,13,11,24,45,65,81)(10,22,43,62,80,36,35)(15,31,44,64,55,70,18)(17,26,46,66,74,87,91)(21,41,60,79,88,52,29)(23,40,59,72,84,69,50)(25,42,61,56,73,86,32)(27,30,49,68,77,85,90)(34,51,63,78,71,53,47)(92,93,94,95,96,97,98,99,100,101,102);(1,3,8,18,36,53,69,83,89,91,90,86,88)(2,27,5,35,7,32,6,47,13,52,17,50,15)(4,14,31,25,23,16,26,10,29,12,11,30,34)(9,51,49,24,20,21,22,46,33,40,42,44,28)(19,64,59,66,41,45,63,48,61,38,43,39,68)(37,60,56,77,74,67,58,72,78,62,55,65,57)

Conjugacy classes & character table

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

Relationships

derived subgroup of	C₁₁⋊#14556975
maximal subgroup of	C₁₁×#14556975, C₁₁×(C₁₃⋊D₁₃)⋊C₇, D₁₁×C₁₃²⋊C₇, …
maximal normal subgroup of	C₁₁×#14556975, C₁₁×(C₁₃⋊D₁₃)⋊C₇, D₁₁×C₁₃²⋊C₇, …
maximal quotient of	C₂×C₁₁×C₁₃²⋊C₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13013, 3);

C11×C132⋊C7 #27189013