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G = C3⋊F13  order 468 = 22·32·13

The semidirect product of C3 and F13 acting via F13/C13⋊C6=C2

metacyclic, supersoluble, monomial, A-group

Aliases: C3⋊F13, C39⋊1C12, C13⋊C6.S3, C39⋊C4⋊C3, C13⋊C3⋊Dic3, C13⋊(C3×Dic3), D13.(C3×S3), (C3×D13).1C6, (C3×C13⋊C3)⋊1C4, (C3×C13⋊C6).1C2, SmallGroup(468,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C39 — C3⋊F13
Chief series
C1 — C13 — C39 — C3×D13 — C3×C13⋊C6 — C3⋊F13
Lower central
C39 — C3⋊F13
Upper central
C1

Generators and relations for C3⋊F13
 G = < a,b,c | a3=b13=c12=1, ab=ba, cac-1=a-1, cbc-1=b6 >

C1
13C2
C3
13C3
26C3
C13
39C4
13C6
13C6
26C6
13C32
D13
C13⋊C3
C39
2C13⋊C3
13Dic3
39C12
13C3×C6
3C13⋊C4
C3×D13
C13⋊C6
2C13⋊C6
C3×C13⋊C3
13C3×Dic3
C39⋊C4
3F13
C3×C13⋊C6
C3⋊F13

Character table of C3⋊F13

 class 123A3B3C3D3E4A4B6A6B6C6D6E12A12B12C12D1339A39B
 size 1132131326263939131326262639393939121212
ρ1111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-1111    linear of order 2
ρ3111ζ3ζ32ζ32ζ3-1-1ζ3ζ32ζ321ζ3ζ65ζ6ζ6ζ65111    linear of order 6
ρ4111ζ32ζ3ζ3ζ32-1-1ζ32ζ3ζ31ζ32ζ6ζ65ζ65ζ6111    linear of order 6
ρ5111ζ3ζ32ζ32ζ311ζ3ζ32ζ321ζ3ζ3ζ32ζ32ζ3111    linear of order 3
ρ6111ζ32ζ3ζ3ζ3211ζ32ζ3ζ31ζ32ζ32ζ3ζ3ζ32111    linear of order 3
ρ71-111111-ii-1-1-1-1-1-ii-ii111    linear of order 4
ρ81-111111i-i-1-1-1-1-1i-ii-i111    linear of order 4
ρ91-11ζ32ζ3ζ3ζ32i-iζ6ζ65ζ65-1ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32111    linear of order 12
ρ101-11ζ32ζ3ζ3ζ32-iiζ6ζ65ζ65-1ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32111    linear of order 12
ρ111-11ζ3ζ32ζ32ζ3-iiζ65ζ6ζ6-1ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3111    linear of order 12
ρ121-11ζ3ζ32ζ32ζ3i-iζ65ζ6ζ6-1ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3111    linear of order 12
ρ1322-122-1-10022-1-1-100002-1-1    orthogonal lifted from S3
ρ142-2-122-1-100-2-211100002-1-1    symplectic lifted from Dic3, Schur index 2
ρ1522-1-1+√-3-1-√-3ζ6ζ6500-1+√-3-1-√-3ζ6-1ζ6500002-1-1    complex lifted from C3×S3
ρ162-2-1-1+√-3-1-√-3ζ6ζ65001-√-31+√-3ζ321ζ300002-1-1    complex lifted from C3×Dic3
ρ1722-1-1-√-3-1+√-3ζ65ζ600-1-√-3-1+√-3ζ65-1ζ600002-1-1    complex lifted from C3×S3
ρ182-2-1-1-√-3-1+√-3ζ65ζ6001+√-31-√-3ζ31ζ3200002-1-1    complex lifted from C3×Dic3
ρ1912012000000000000000-1-1-1    orthogonal lifted from F13
ρ20120-6000000000000000-11-√-39/21+√-39/2    complex faithful
ρ21120-6000000000000000-11+√-39/21-√-39/2    complex faithful

Smallest permutation representation of C3⋊F13
►On 39 points
Generators in S39
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)

(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)

(2 12 5 6 4 8 13 3 10 9 11 7)(14 27)(15 38 18 32 17 34 26 29 23 35 24 33)(16 36 22 37 20 28 25 31 19 30 21 39)
 
G:=sub<Sym(39)| (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26), (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), (2,12,5,6,4,8,13,3,10,9,11,7)(14,27)(15,38,18,32,17,34,26,29,23,35,24,33)(16,36,22,37,20,28,25,31,19,30,21,39)>;
 
G:=Group( (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26), (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), (2,12,5,6,4,8,13,3,10,9,11,7)(14,27)(15,38,18,32,17,34,26,29,23,35,24,33)(16,36,22,37,20,28,25,31,19,30,21,39) );
 
G=PermutationGroup([[(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26)], [(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)], [(2,12,5,6,4,8,13,3,10,9,11,7),(14,27),(15,38,18,32,17,34,26,29,23,35,24,33),(16,36,22,37,20,28,25,31,19,30,21,39)]])
 

Matrix representation of C3⋊F13 ►in GL14(𝔽157)

113000000000000
36155000000000000
00100000000000
00010000000000
00001000000000
00000100000000
00000010000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
10000000000000
01000000000000
00156156156156156156156156156156156156
00100000000000
00010000000000
00001000000000
00000100000000
00000010000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
,
10000000000000
36156000000000000
00100000000000
00000000100000
00156156156156156156156156156156156156
00000001000000
00000000000001
00000010000000
00000000000010
00000100000000
00000000000100
00001000000000
00000000001000
00010000000000

G:=sub<GL(14,GF(157))| [1,36,0,0,0,0,0,0,0,0,0,0,0,0,13,155,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,156,1,0,0,0,0,0,0,0,0,0,0,0,0,156,0,1,0,0,0,0,0,0,0,0,0,0,0,156,0,0,1,0,0,0,0,0,0,0,0,0,0,156,0,0,0,1,0,0,0,0,0,0,0,0,0,156,0,0,0,0,1,0,0,0,0,0,0,0,0,156,0,0,0,0,0,1,0,0,0,0,0,0,0,156,0,0,0,0,0,0,1,0,0,0,0,0,0,156,0,0,0,0,0,0,0,1,0,0,0,0,0,156,0,0,0,0,0,0,0,0,1,0,0,0,0,156,0,0,0,0,0,0,0,0,0,1,0,0,0,156,0,0,0,0,0,0,0,0,0,0,1,0,0,156,0,0,0,0,0,0,0,0,0,0,0],[1,36,0,0,0,0,0,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,156,0,0,0,0,0,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,0,1,0,0,0,0,156,0,0,0,0,0,0,1,0,0,0,0,0,0,156,0,0,0,0,1,0,0,0,0,0,0,0,0,156,0,0,1,0,0,0,0,0,0,0,0,0,0,156,1,0,0,0,0,0,0,0,0,0,0,0,1,156,0,0,0,0,0,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,1,0,0,0,0,0,156,0,0,0,0,0,1,0,0,0,0,0,0,0,156,0,0,0,1,0,0,0,0,0,0,0,0,0,156,0,1,0,0,0,0,0,0,0] >;
 

C3⋊F13 in GAP, Magma, Sage, TeX 

C_3\rtimes F_{13}
 
% in TeX
 
G:=Group("C3:F13");
 
// GroupNames label
 
G:=SmallGroup(468,30);
 
// by ID
 
G=gap.SmallGroup(468,30);
 
# by ID
 
G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,7204,4059,1814]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^13=c^12=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C3⋊F13 in TeX
Character table of C3⋊F13 in TeX

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