direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C4×C13⋊C3, C52⋊4C6, C26⋊4C12, (C2×C52)⋊C3, C13⋊5(C2×C12), C26.6(C2×C6), (C2×C26).2C6, C22.(C2×C13⋊C3), C2.1(C22×C13⋊C3), (C22×C13⋊C3).2C2, (C2×C13⋊C3).6C22, SmallGroup(312,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C13 — C26 — C2×C13⋊C3 — C22×C13⋊C3 — C2×C4×C13⋊C3 |
| C13 — C2×C4×C13⋊C3 |
Generators and relations for C2×C4×C13⋊C3
G = < a,b,c,d | a2=b4=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Smallest permutation representation of C2×C4×C13⋊C3
►On 104 points
Generators in S104
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)
(1 27 14 40)(2 28 15 41)(3 29 16 42)(4 30 17 43)(5 31 18 44)(6 32 19 45)(7 33 20 46)(8 34 21 47)(9 35 22 48)(10 36 23 49)(11 37 24 50)(12 38 25 51)(13 39 26 52)(53 79 66 92)(54 80 67 93)(55 81 68 94)(56 82 69 95)(57 83 70 96)(58 84 71 97)(59 85 72 98)(60 86 73 99)(61 87 74 100)(62 88 75 101)(63 89 76 102)(64 90 77 103)(65 91 78 104)
(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)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91), (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52)(53,79,66,92)(54,80,67,93)(55,81,68,94)(56,82,69,95)(57,83,70,96)(58,84,71,97)(59,85,72,98)(60,86,73,99)(61,87,74,100)(62,88,75,101)(63,89,76,102)(64,90,77,103)(65,91,78,104), (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( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91), (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52)(53,79,66,92)(54,80,67,93)(55,81,68,94)(56,82,69,95)(57,83,70,96)(58,84,71,97)(59,85,72,98)(60,86,73,99)(61,87,74,100)(62,88,75,101)(63,89,76,102)(64,90,77,103)(65,91,78,104), (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([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91)], [(1,27,14,40),(2,28,15,41),(3,29,16,42),(4,30,17,43),(5,31,18,44),(6,32,19,45),(7,33,20,46),(8,34,21,47),(9,35,22,48),(10,36,23,49),(11,37,24,50),(12,38,25,51),(13,39,26,52),(53,79,66,92),(54,80,67,93),(55,81,68,94),(56,82,69,95),(57,83,70,96),(58,84,71,97),(59,85,72,98),(60,86,73,99),(61,87,74,100),(62,88,75,101),(63,89,76,102),(64,90,77,103),(65,91,78,104)], [(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 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 13A | 13B | 13C | 13D | 26A | ··· | 26L | 52A | ··· | 52P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 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 | C2 | C3 | C4 | C6 | C6 | C12 | C13⋊C3 | C2×C13⋊C3 | C2×C13⋊C3 | C4×C13⋊C3 |
| kernel | C2×C4×C13⋊C3 | C4×C13⋊C3 | C22×C13⋊C3 | C2×C52 | C2×C13⋊C3 | C52 | C2×C26 | C26 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 4 | 16 |
Matrix representation of C2×C4×C13⋊C3 ►in GL4(𝔽157) generated by
| 156 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 156 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 52 |
| 0 | 0 | 1 | 119 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 52 |
| 0 | 0 | 52 | 118 |
| 0 | 0 | 119 | 104 |
G:=sub<GL(4,GF(157))| [156,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[156,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,52,119],[12,0,0,0,0,1,0,0,0,1,52,119,0,52,118,104] >;
C2×C4×C13⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{13}\rtimes C_3 % in TeX
G:=Group("C2xC4xC13:C3"); // GroupNames label
G:=SmallGroup(312,22);
// by ID
G=gap.SmallGroup(312,22);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,66,464]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^13=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
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