direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C8×C13⋊C3, C104⋊C3, C13⋊4C24, C52.4C6, C26.4C12, C2.(C4×C13⋊C3), C4.2(C2×C13⋊C3), (C4×C13⋊C3).4C2, (C2×C13⋊C3).3C4, SmallGroup(312,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C8×C13⋊C3 |
Generators and relations for C8×C13⋊C3
G = < a,b,c | a8=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C8×C13⋊C3
►On 104 points
Generators in S104
(1 92 40 66 14 79 27 53)(2 93 41 67 15 80 28 54)(3 94 42 68 16 81 29 55)(4 95 43 69 17 82 30 56)(5 96 44 70 18 83 31 57)(6 97 45 71 19 84 32 58)(7 98 46 72 20 85 33 59)(8 99 47 73 21 86 34 60)(9 100 48 74 22 87 35 61)(10 101 49 75 23 88 36 62)(11 102 50 76 24 89 37 63)(12 103 51 77 25 90 38 64)(13 104 52 78 26 91 39 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)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)(93 95 101)(94 98 97)(96 104 102)(99 100 103)
G:=sub<Sym(104)| (1,92,40,66,14,79,27,53)(2,93,41,67,15,80,28,54)(3,94,42,68,16,81,29,55)(4,95,43,69,17,82,30,56)(5,96,44,70,18,83,31,57)(6,97,45,71,19,84,32,58)(7,98,46,72,20,85,33,59)(8,99,47,73,21,86,34,60)(9,100,48,74,22,87,35,61)(10,101,49,75,23,88,36,62)(11,102,50,76,24,89,37,63)(12,103,51,77,25,90,38,64)(13,104,52,78,26,91,39,65), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103)>;
G:=Group( (1,92,40,66,14,79,27,53)(2,93,41,67,15,80,28,54)(3,94,42,68,16,81,29,55)(4,95,43,69,17,82,30,56)(5,96,44,70,18,83,31,57)(6,97,45,71,19,84,32,58)(7,98,46,72,20,85,33,59)(8,99,47,73,21,86,34,60)(9,100,48,74,22,87,35,61)(10,101,49,75,23,88,36,62)(11,102,50,76,24,89,37,63)(12,103,51,77,25,90,38,64)(13,104,52,78,26,91,39,65), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103) );
G=PermutationGroup([[(1,92,40,66,14,79,27,53),(2,93,41,67,15,80,28,54),(3,94,42,68,16,81,29,55),(4,95,43,69,17,82,30,56),(5,96,44,70,18,83,31,57),(6,97,45,71,19,84,32,58),(7,98,46,72,20,85,33,59),(8,99,47,73,21,86,34,60),(9,100,48,74,22,87,35,61),(10,101,49,75,23,88,36,62),(11,102,50,76,24,89,37,63),(12,103,51,77,25,90,38,64),(13,104,52,78,26,91,39,65)], [(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),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(67,69,75),(68,72,71),(70,78,76),(73,74,77),(80,82,88),(81,85,84),(83,91,89),(86,87,90),(93,95,101),(94,98,97),(96,104,102),(99,100,103)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 13A | 13B | 13C | 13D | 24A | ··· | 24H | 26A | 26B | 26C | 26D | 52A | ··· | 52H | 104A | ··· | 104P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 13 | 13 | 13 | 13 | 24 | ··· | 24 | 26 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C13⋊C3 | C2×C13⋊C3 | C4×C13⋊C3 | C8×C13⋊C3 |
| kernel | C8×C13⋊C3 | C4×C13⋊C3 | C104 | C2×C13⋊C3 | C52 | C13⋊C3 | C26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 |
Matrix representation of C8×C13⋊C3 ►in GL4(𝔽313) generated by
| 188 | 0 | 0 | 0 |
| 0 | 288 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 0 | 288 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 118 |
| 0 | 0 | 1 | 7 |
| 214 | 0 | 0 | 0 |
| 0 | 1 | 1 | 118 |
| 0 | 0 | 118 | 6 |
| 0 | 0 | 7 | 194 |
G:=sub<GL(4,GF(313))| [188,0,0,0,0,288,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,118,7],[214,0,0,0,0,1,0,0,0,1,118,7,0,118,6,194] >;
C8×C13⋊C3 in GAP, Magma, Sage, TeX
C_8\times C_{13}\rtimes C_3 % in TeX
G:=Group("C8xC13:C3"); // GroupNames label
G:=SmallGroup(312,2);
// by ID
G=gap.SmallGroup(312,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,909]);
// Polycyclic
G:=Group<a,b,c|a^8=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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