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G = C6×C13⋊C4order 312 = 23·3·13

Direct product of C6 and C13⋊C4

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C6×C13⋊C4, C782C4, C263C12, D133C12, D26.3C6, C393(C2×C4), C134(C2×C12), (C3×D13)⋊3C4, D13.2(C2×C6), (C6×D13).3C2, (C3×D13).3C22, SmallGroup(312,52)

Series: Derived Chief Lower central Upper central

C1C13 — C6×C13⋊C4
C1C13D13C3×D13C3×C13⋊C4 — C6×C13⋊C4
C13 — C6×C13⋊C4
C1C6

Generators and relations for C6×C13⋊C4
 G = < a,b,c | a6=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C2
13C4
13C22
13C4
13C6
13C6
13C2×C4
13C12
13C12
13C2×C6
13C2×C12

Smallest permutation representation of C6×C13⋊C4
On 78 points
Generators in S78
(1 53 27 40 14 66)(2 54 28 41 15 67)(3 55 29 42 16 68)(4 56 30 43 17 69)(5 57 31 44 18 70)(6 58 32 45 19 71)(7 59 33 46 20 72)(8 60 34 47 21 73)(9 61 35 48 22 74)(10 62 36 49 23 75)(11 63 37 50 24 76)(12 64 38 51 25 77)(13 65 39 52 26 78)
(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)(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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)(67 74 78 71)(68 69 77 76)(70 72 75 73)

G:=sub<Sym(78)| (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78), (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)(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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73)>;

G:=Group( (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78), (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)(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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73) );

G=PermutationGroup([[(1,53,27,40,14,66),(2,54,28,41,15,67),(3,55,29,42,16,68),(4,56,30,43,17,69),(5,57,31,44,18,70),(6,58,32,45,19,71),(7,59,33,46,20,72),(8,60,34,47,21,73),(9,61,35,48,22,74),(10,62,36,49,23,75),(11,63,37,50,24,76),(12,64,38,51,25,77),(13,65,39,52,26,78)], [(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),(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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34),(41,48,52,45),(42,43,51,50),(44,46,49,47),(54,61,65,58),(55,56,64,63),(57,59,62,60),(67,74,78,71),(68,69,77,76),(70,72,75,73)]])

42 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A···12H13A13B13C26A26B26C39A···39F78A···78F
order122233444466666612···1213131326262639···3978···78
size1113131113131313111313131313···134444444···44···4

42 irreducible representations

dim11111111114444
type+++++
imageC1C2C2C3C4C4C6C6C12C12C13⋊C4C2×C13⋊C4C3×C13⋊C4C6×C13⋊C4
kernelC6×C13⋊C4C3×C13⋊C4C6×D13C2×C13⋊C4C3×D13C78C13⋊C4D26D13C26C6C3C2C1
# reps12122242443366

Matrix representation of C6×C13⋊C4 in GL5(𝔽157)

1450000
01000
00100
00010
00001
,
10000
0205220156
01000
00100
00010
,
1560000
01000
013785118138
052397119
00010

G:=sub<GL(5,GF(157))| [145,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,20,1,0,0,0,52,0,1,0,0,20,0,0,1,0,156,0,0,0],[156,0,0,0,0,0,1,137,52,0,0,0,85,39,0,0,0,118,71,1,0,0,138,19,0] >;

C6×C13⋊C4 in GAP, Magma, Sage, TeX

C_6\times C_{13}\rtimes C_4
% in TeX

G:=Group("C6xC13:C4");
// GroupNames label

G:=SmallGroup(312,52);
// by ID

G=gap.SmallGroup(312,52);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,4804,619]);
// Polycyclic

G:=Group<a,b,c|a^6=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C6×C13⋊C4 in TeX

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