Copied to
clipboard

## G = C6×C13⋊C6order 468 = 22·32·13

### Direct product of C6 and C13⋊C6

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

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

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 2A 2B 2C 3A 3B 3C ··· 3H 6A 6B 6C ··· 6X 13A 13B 26A 26B 39A 39B 39C 39D 78A 78B 78C 78D order 1 2 2 2 3 3 3 ··· 3 6 6 6 ··· 6 13 13 26 26 39 39 39 39 78 78 78 78 size 1 1 13 13 1 1 13 ··· 13 1 1 13 ··· 13 6 6 6 6 6 6 6 6 6 6 6 6

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 C13⋊C6 C2×C13⋊C6 C3×C13⋊C6 C6×C13⋊C6 kernel C6×C13⋊C6 C3×C13⋊C6 C6×C13⋊C3 C2×C13⋊C6 C6×D13 C13⋊C6 C2×C13⋊C3 C3×D13 C78 C6 C3 C2 C1 # reps 1 2 1 6 2 12 6 4 2 2 2 4 4

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

 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 77 16 77 15 78 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 56 0 0 0 0 0 0 0 78 0 0 0 0 0 0 66 50 2 64 67 65 0 15 13 15 78 14 14 0 0 0 0 0 78 0 0 0 78 0 0 0 0 0 14 14 78 15 13 15

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

׿
×
𝔽