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G = C6×C13⋊C6order 468 = 22·32·13

Direct product of C6 and C13⋊C6

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

Aliases: C6×C13⋊C6, C13⋊C62, C782C6, D26⋊C32, C26⋊(C3×C6), D13⋊(C3×C6), (C6×D13)⋊C3, C393(C2×C6), (C3×D13)⋊2C6, (C2×C13⋊C3)⋊C6, C13⋊C3⋊(C2×C6), (C6×C13⋊C3)⋊2C2, (C3×C13⋊C3)⋊3C22, SmallGroup(468,33)

Series: Derived Chief Lower central Upper central

C1C13 — C6×C13⋊C6
C1C13C39C3×C13⋊C3C3×C13⋊C6 — C6×C13⋊C6
C13 — C6×C13⋊C6
C1C6

Generators and relations for C6×C13⋊C6
 G = < a,b,c | a6=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >

Subgroups: 372 in 60 conjugacy classes, 34 normal (13 characteristic)
C1, C2, C2, C3, C3, C22, C6, C6, C32, C2×C6, C13, C3×C6, D13, C26, C62, C13⋊C3, C39, D26, C13⋊C6, C2×C13⋊C3, C3×D13, C78, C3×C13⋊C3, C2×C13⋊C6, C6×D13, C3×C13⋊C6, C6×C13⋊C3, C6×C13⋊C6
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C62, C13⋊C6, C2×C13⋊C6, C3×C13⋊C6, C6×C13⋊C6

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

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

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C···3H6A6B6C···6X13A13B26A26B39A39B39C39D78A78B78C78D
order1222333···3666···6131326263939393978787878
size1113131113···131113···13666666666666

48 irreducible representations

dim1111111116666
type+++++
imageC1C2C2C3C3C6C6C6C6C13⋊C6C2×C13⋊C6C3×C13⋊C6C6×C13⋊C6
kernelC6×C13⋊C6C3×C13⋊C6C6×C13⋊C3C2×C13⋊C6C6×D13C13⋊C6C2×C13⋊C3C3×D13C78C6C3C2C1
# reps12162126422244

Matrix representation of C6×C13⋊C6 in GL7(𝔽79)

23000000
07800000
00780000
00078000
00007800
00000780
00000078
,
1000000
0157716771578
0100000
0010000
0001000
0000100
0000010
,
56000000
07800000
066502646765
0151315781414
00000780
00780000
0141478151315

G:=sub<GL(7,GF(79))| [23,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78,0,0,0,0,0,0,0,78],[1,0,0,0,0,0,0,0,15,1,0,0,0,0,0,77,0,1,0,0,0,0,16,0,0,1,0,0,0,77,0,0,0,1,0,0,15,0,0,0,0,1,0,78,0,0,0,0,0],[56,0,0,0,0,0,0,0,78,66,15,0,0,14,0,0,50,13,0,78,14,0,0,2,15,0,0,78,0,0,64,78,0,0,15,0,0,67,14,78,0,13,0,0,65,14,0,0,15] >;

C6×C13⋊C6 in GAP, Magma, Sage, TeX

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

G:=Group("C6xC13:C6");
// GroupNames label

G:=SmallGroup(468,33);
// by ID

G=gap.SmallGroup(468,33);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,10804,689]);
// Polycyclic

G:=Group<a,b,c|a^6=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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