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## G = C6×C13⋊C3order 234 = 2·32·13

### Direct product of C6 and C13⋊C3

Aliases: C6×C13⋊C3, C78⋊C3, C26⋊C32, C394C6, C132(C3×C6), SmallGroup(234,10)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C13⋊C3
G = < a,b,c | a6=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Smallest permutation representation of C6×C13⋊C3
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 27 14)(2 30 23)(3 33 19)(4 36 15)(5 39 24)(6 29 20)(7 32 16)(8 35 25)(9 38 21)(10 28 17)(11 31 26)(12 34 22)(13 37 18)(40 66 53)(41 69 62)(42 72 58)(43 75 54)(44 78 63)(45 68 59)(46 71 55)(47 74 64)(48 77 60)(49 67 56)(50 70 65)(51 73 61)(52 76 57)

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

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

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

C6×C13⋊C3 is a maximal subgroup of   C393C12

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C13⋊C3 C2×C13⋊C3 C3×C13⋊C3 C6×C13⋊C3 kernel C6×C13⋊C3 C3×C13⋊C3 C2×C13⋊C3 C78 C13⋊C3 C39 C6 C3 C2 C1 # reps 1 1 6 2 6 2 4 4 8 8

Matrix representation of C6×C13⋊C3 in GL3(𝔽79) generated by

 24 0 0 0 24 0 0 0 24
,
 68 4 1 4 42 50 56 74 38
,
 15 12 3 64 8 49 64 43 56
G:=sub<GL(3,GF(79))| [24,0,0,0,24,0,0,0,24],[68,4,56,4,42,74,1,50,38],[15,64,64,12,8,43,3,49,56] >;

C6×C13⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_{13}\rtimes C_3
% in TeX

G:=Group("C6xC13:C3");
// GroupNames label

G:=SmallGroup(234,10);
// by ID

G=gap.SmallGroup(234,10);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,439]);
// Polycyclic

G:=Group<a,b,c|a^6=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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