direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C39⋊C4, C39⋊6C12, C39⋊2Dic3, (C3×C39)⋊3C4, C32⋊2(C13⋊C4), C13⋊3(C3×Dic3), (C3×D13).4S3, (C3×D13).5C6, D13.2(C3×S3), (C32×D13).2C2, C3⋊(C3×C13⋊C4), SmallGroup(468,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C3×C39⋊C4 |
Generators and relations for C3×C39⋊C4
G = < a,b,c | a3=b39=c4=1, ab=ba, ac=ca, cbc-1=b8 >
Smallest permutation representation of C3×C39⋊C4
►On 78 points
Generators in S78
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(40 66 53)(41 67 54)(42 68 55)(43 69 56)(44 70 57)(45 71 58)(46 72 59)(47 73 60)(48 74 61)(49 75 62)(50 76 63)(51 77 64)(52 78 65)
(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 49)(2 54 26 57)(3 59 12 65)(4 64 37 73)(5 69 23 42)(6 74 9 50)(7 40 34 58)(8 45 20 66)(10 55 31 43)(11 60 17 51)(13 70 28 67)(14 75)(15 41 39 44)(16 46 25 52)(18 56 36 68)(19 61 22 76)(21 71 33 53)(24 47 30 77)(27 62)(29 72 38 78)(32 48 35 63)
G:=sub<Sym(78)| (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(40,66,53)(41,67,54)(42,68,55)(43,69,56)(44,70,57)(45,71,58)(46,72,59)(47,73,60)(48,74,61)(49,75,62)(50,76,63)(51,77,64)(52,78,65), (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), (1,49)(2,54,26,57)(3,59,12,65)(4,64,37,73)(5,69,23,42)(6,74,9,50)(7,40,34,58)(8,45,20,66)(10,55,31,43)(11,60,17,51)(13,70,28,67)(14,75)(15,41,39,44)(16,46,25,52)(18,56,36,68)(19,61,22,76)(21,71,33,53)(24,47,30,77)(27,62)(29,72,38,78)(32,48,35,63)>;
G:=Group( (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(40,66,53)(41,67,54)(42,68,55)(43,69,56)(44,70,57)(45,71,58)(46,72,59)(47,73,60)(48,74,61)(49,75,62)(50,76,63)(51,77,64)(52,78,65), (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), (1,49)(2,54,26,57)(3,59,12,65)(4,64,37,73)(5,69,23,42)(6,74,9,50)(7,40,34,58)(8,45,20,66)(10,55,31,43)(11,60,17,51)(13,70,28,67)(14,75)(15,41,39,44)(16,46,25,52)(18,56,36,68)(19,61,22,76)(21,71,33,53)(24,47,30,77)(27,62)(29,72,38,78)(32,48,35,63) );
G=PermutationGroup([[(1,14,27),(2,15,28),(3,16,29),(4,17,30),(5,18,31),(6,19,32),(7,20,33),(8,21,34),(9,22,35),(10,23,36),(11,24,37),(12,25,38),(13,26,39),(40,66,53),(41,67,54),(42,68,55),(43,69,56),(44,70,57),(45,71,58),(46,72,59),(47,73,60),(48,74,61),(49,75,62),(50,76,63),(51,77,64),(52,78,65)], [(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)], [(1,49),(2,54,26,57),(3,59,12,65),(4,64,37,73),(5,69,23,42),(6,74,9,50),(7,40,34,58),(8,45,20,66),(10,55,31,43),(11,60,17,51),(13,70,28,67),(14,75),(15,41,39,44),(16,46,25,52),(18,56,36,68),(19,61,22,76),(21,71,33,53),(24,47,30,77),(27,62),(29,72,38,78),(32,48,35,63)]])
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 13A | 13B | 13C | 39A | ··· | 39X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 13 | 1 | 1 | 2 | 2 | 2 | 39 | 39 | 13 | 13 | 26 | 26 | 26 | 39 | 39 | 39 | 39 | 4 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | C13⋊C4 | C3×C13⋊C4 | C39⋊C4 | C3×C39⋊C4 |
| kernel | C3×C39⋊C4 | C32×D13 | C39⋊C4 | C3×C39 | C3×D13 | C39 | C3×D13 | C39 | D13 | C13 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 3 | 6 | 6 | 12 |
Matrix representation of C3×C39⋊C4 ►in GL4(𝔽157) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 138 | 52 | 0 | 0 |
| 105 | 143 | 0 | 0 |
| 0 | 0 | 145 | 145 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 142 | 156 | 0 | 0 |
G:=sub<GL(4,GF(157))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[138,105,0,0,52,143,0,0,0,0,145,12,0,0,145,11],[0,0,1,142,0,0,0,156,1,0,0,0,0,1,0,0] >;
C3×C39⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{39}\rtimes C_4 % in TeX
G:=Group("C3xC39:C4"); // GroupNames label
G:=SmallGroup(468,37);
// by ID
G=gap.SmallGroup(468,37);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,4504,1814]);
// Polycyclic
G:=Group<a,b,c|a^3=b^39=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^8>;
// generators/relations
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