direct product, non-abelian, soluble
Aliases: C3×2- 1+4⋊C5, 2- 1+4⋊C15, C6.(C24⋊C5), (C3×2- 1+4)⋊C5, C2.(C3×C24⋊C5), SmallGroup(480,1046)
Series: Derived ►Chief ►Lower central ►Upper central
| 2- 1+4 — C3×2- 1+4⋊C5 |
Generators and relations for C3×2- 1+4⋊C5
G = < a,b,c,d,e,f | a3=b4=c2=f5=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2cde, cd=dc, ce=ec, fcf-1=bc, ede-1=b2d, fdf-1=bcd, fef-1=de >
Smallest permutation representation of C3×2- 1+4⋊C5
►On 96 points
Generators in S96
(1 5 2)(3 4 6)(7 89 73)(8 90 74)(9 91 75)(10 87 76)(11 88 72)(12 61 41)(13 57 37)(14 58 38)(15 59 39)(16 60 40)(17 63 82)(18 64 83)(19 65 84)(20 66 85)(21 62 86)(22 43 69)(23 44 70)(24 45 71)(25 46 67)(26 42 68)(27 52 47)(28 53 48)(29 54 49)(30 55 50)(31 56 51)(32 96 77)(33 92 78)(34 93 79)(35 94 80)(36 95 81)
(1 83 3 95)(2 64 6 36)(4 81 5 18)(7 65 37 32)(8 10 38 40)(9 51 39 67)(11 35 41 63)(12 82 88 94)(13 96 89 84)(14 16 90 87)(15 25 91 31)(17 72 80 61)(19 57 77 73)(20 42 78 52)(21 45 79 55)(22 23 28 29)(24 93 30 86)(26 92 27 85)(33 47 66 68)(34 50 62 71)(43 44 53 54)(46 75 56 59)(48 49 69 70)(58 60 74 76)
(1 23)(2 70)(3 29)(4 54)(5 44)(6 49)(7 62)(8 51)(9 10)(11 68)(12 27)(13 93)(14 25)(15 16)(17 78)(18 43)(19 55)(20 80)(21 73)(22 83)(24 96)(26 88)(28 95)(30 84)(31 90)(32 71)(33 63)(34 37)(35 66)(36 48)(38 67)(39 40)(41 47)(42 72)(45 77)(46 58)(50 65)(52 61)(53 81)(56 74)(57 79)(59 60)(64 69)(75 76)(82 92)(85 94)(86 89)(87 91)
(1 15 3 91)(2 39 6 9)(4 75 5 59)(7 63 37 35)(8 69 38 48)(10 70 40 49)(11 32 41 65)(12 84 88 96)(13 94 89 82)(14 28 90 22)(16 29 87 23)(17 57 80 73)(18 46 81 56)(19 72 77 61)(20 21 78 79)(24 27 30 26)(25 95 31 83)(33 34 66 62)(36 51 64 67)(42 45 52 55)(43 58 53 74)(44 60 54 76)(47 50 68 71)(85 86 92 93)
(1 86 3 93)(2 62 6 34)(4 79 5 21)(7 49 37 70)(8 41 38 11)(9 33 39 66)(10 63 40 35)(12 14 88 90)(13 23 89 29)(15 85 91 92)(16 94 87 82)(17 60 80 76)(18 45 81 55)(19 43 77 53)(20 75 78 59)(22 96 28 84)(24 95 30 83)(25 26 31 27)(32 48 65 69)(36 50 64 71)(42 56 52 46)(44 73 54 57)(47 67 68 51)(58 72 74 61)
(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)
G:=sub<Sym(96)| (1,5,2)(3,4,6)(7,89,73)(8,90,74)(9,91,75)(10,87,76)(11,88,72)(12,61,41)(13,57,37)(14,58,38)(15,59,39)(16,60,40)(17,63,82)(18,64,83)(19,65,84)(20,66,85)(21,62,86)(22,43,69)(23,44,70)(24,45,71)(25,46,67)(26,42,68)(27,52,47)(28,53,48)(29,54,49)(30,55,50)(31,56,51)(32,96,77)(33,92,78)(34,93,79)(35,94,80)(36,95,81), (1,83,3,95)(2,64,6,36)(4,81,5,18)(7,65,37,32)(8,10,38,40)(9,51,39,67)(11,35,41,63)(12,82,88,94)(13,96,89,84)(14,16,90,87)(15,25,91,31)(17,72,80,61)(19,57,77,73)(20,42,78,52)(21,45,79,55)(22,23,28,29)(24,93,30,86)(26,92,27,85)(33,47,66,68)(34,50,62,71)(43,44,53,54)(46,75,56,59)(48,49,69,70)(58,60,74,76), (1,23)(2,70)(3,29)(4,54)(5,44)(6,49)(7,62)(8,51)(9,10)(11,68)(12,27)(13,93)(14,25)(15,16)(17,78)(18,43)(19,55)(20,80)(21,73)(22,83)(24,96)(26,88)(28,95)(30,84)(31,90)(32,71)(33,63)(34,37)(35,66)(36,48)(38,67)(39,40)(41,47)(42,72)(45,77)(46,58)(50,65)(52,61)(53,81)(56,74)(57,79)(59,60)(64,69)(75,76)(82,92)(85,94)(86,89)(87,91), (1,15,3,91)(2,39,6,9)(4,75,5,59)(7,63,37,35)(8,69,38,48)(10,70,40,49)(11,32,41,65)(12,84,88,96)(13,94,89,82)(14,28,90,22)(16,29,87,23)(17,57,80,73)(18,46,81,56)(19,72,77,61)(20,21,78,79)(24,27,30,26)(25,95,31,83)(33,34,66,62)(36,51,64,67)(42,45,52,55)(43,58,53,74)(44,60,54,76)(47,50,68,71)(85,86,92,93), (1,86,3,93)(2,62,6,34)(4,79,5,21)(7,49,37,70)(8,41,38,11)(9,33,39,66)(10,63,40,35)(12,14,88,90)(13,23,89,29)(15,85,91,92)(16,94,87,82)(17,60,80,76)(18,45,81,55)(19,43,77,53)(20,75,78,59)(22,96,28,84)(24,95,30,83)(25,26,31,27)(32,48,65,69)(36,50,64,71)(42,56,52,46)(44,73,54,57)(47,67,68,51)(58,72,74,61), (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)>;
G:=Group( (1,5,2)(3,4,6)(7,89,73)(8,90,74)(9,91,75)(10,87,76)(11,88,72)(12,61,41)(13,57,37)(14,58,38)(15,59,39)(16,60,40)(17,63,82)(18,64,83)(19,65,84)(20,66,85)(21,62,86)(22,43,69)(23,44,70)(24,45,71)(25,46,67)(26,42,68)(27,52,47)(28,53,48)(29,54,49)(30,55,50)(31,56,51)(32,96,77)(33,92,78)(34,93,79)(35,94,80)(36,95,81), (1,83,3,95)(2,64,6,36)(4,81,5,18)(7,65,37,32)(8,10,38,40)(9,51,39,67)(11,35,41,63)(12,82,88,94)(13,96,89,84)(14,16,90,87)(15,25,91,31)(17,72,80,61)(19,57,77,73)(20,42,78,52)(21,45,79,55)(22,23,28,29)(24,93,30,86)(26,92,27,85)(33,47,66,68)(34,50,62,71)(43,44,53,54)(46,75,56,59)(48,49,69,70)(58,60,74,76), (1,23)(2,70)(3,29)(4,54)(5,44)(6,49)(7,62)(8,51)(9,10)(11,68)(12,27)(13,93)(14,25)(15,16)(17,78)(18,43)(19,55)(20,80)(21,73)(22,83)(24,96)(26,88)(28,95)(30,84)(31,90)(32,71)(33,63)(34,37)(35,66)(36,48)(38,67)(39,40)(41,47)(42,72)(45,77)(46,58)(50,65)(52,61)(53,81)(56,74)(57,79)(59,60)(64,69)(75,76)(82,92)(85,94)(86,89)(87,91), (1,15,3,91)(2,39,6,9)(4,75,5,59)(7,63,37,35)(8,69,38,48)(10,70,40,49)(11,32,41,65)(12,84,88,96)(13,94,89,82)(14,28,90,22)(16,29,87,23)(17,57,80,73)(18,46,81,56)(19,72,77,61)(20,21,78,79)(24,27,30,26)(25,95,31,83)(33,34,66,62)(36,51,64,67)(42,45,52,55)(43,58,53,74)(44,60,54,76)(47,50,68,71)(85,86,92,93), (1,86,3,93)(2,62,6,34)(4,79,5,21)(7,49,37,70)(8,41,38,11)(9,33,39,66)(10,63,40,35)(12,14,88,90)(13,23,89,29)(15,85,91,92)(16,94,87,82)(17,60,80,76)(18,45,81,55)(19,43,77,53)(20,75,78,59)(22,96,28,84)(24,95,30,83)(25,26,31,27)(32,48,65,69)(36,50,64,71)(42,56,52,46)(44,73,54,57)(47,67,68,51)(58,72,74,61), (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) );
G=PermutationGroup([[(1,5,2),(3,4,6),(7,89,73),(8,90,74),(9,91,75),(10,87,76),(11,88,72),(12,61,41),(13,57,37),(14,58,38),(15,59,39),(16,60,40),(17,63,82),(18,64,83),(19,65,84),(20,66,85),(21,62,86),(22,43,69),(23,44,70),(24,45,71),(25,46,67),(26,42,68),(27,52,47),(28,53,48),(29,54,49),(30,55,50),(31,56,51),(32,96,77),(33,92,78),(34,93,79),(35,94,80),(36,95,81)], [(1,83,3,95),(2,64,6,36),(4,81,5,18),(7,65,37,32),(8,10,38,40),(9,51,39,67),(11,35,41,63),(12,82,88,94),(13,96,89,84),(14,16,90,87),(15,25,91,31),(17,72,80,61),(19,57,77,73),(20,42,78,52),(21,45,79,55),(22,23,28,29),(24,93,30,86),(26,92,27,85),(33,47,66,68),(34,50,62,71),(43,44,53,54),(46,75,56,59),(48,49,69,70),(58,60,74,76)], [(1,23),(2,70),(3,29),(4,54),(5,44),(6,49),(7,62),(8,51),(9,10),(11,68),(12,27),(13,93),(14,25),(15,16),(17,78),(18,43),(19,55),(20,80),(21,73),(22,83),(24,96),(26,88),(28,95),(30,84),(31,90),(32,71),(33,63),(34,37),(35,66),(36,48),(38,67),(39,40),(41,47),(42,72),(45,77),(46,58),(50,65),(52,61),(53,81),(56,74),(57,79),(59,60),(64,69),(75,76),(82,92),(85,94),(86,89),(87,91)], [(1,15,3,91),(2,39,6,9),(4,75,5,59),(7,63,37,35),(8,69,38,48),(10,70,40,49),(11,32,41,65),(12,84,88,96),(13,94,89,82),(14,28,90,22),(16,29,87,23),(17,57,80,73),(18,46,81,56),(19,72,77,61),(20,21,78,79),(24,27,30,26),(25,95,31,83),(33,34,66,62),(36,51,64,67),(42,45,52,55),(43,58,53,74),(44,60,54,76),(47,50,68,71),(85,86,92,93)], [(1,86,3,93),(2,62,6,34),(4,79,5,21),(7,49,37,70),(8,41,38,11),(9,33,39,66),(10,63,40,35),(12,14,88,90),(13,23,89,29),(15,85,91,92),(16,94,87,82),(17,60,80,76),(18,45,81,55),(19,43,77,53),(20,75,78,59),(22,96,28,84),(24,95,30,83),(25,26,31,27),(32,48,65,69),(36,50,64,71),(42,56,52,46),(44,73,54,57),(47,67,68,51),(58,72,74,61)], [(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)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 10 | 16 | 16 | 16 | 16 | 1 | 1 | 10 | 10 | 16 | 16 | 16 | 16 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 16 | ··· | 16 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 5 | 5 |
| type | + | - | + | ||||||
| image | C1 | C3 | C5 | C15 | 2- 1+4⋊C5 | 2- 1+4⋊C5 | C3×2- 1+4⋊C5 | C24⋊C5 | C3×C24⋊C5 |
| kernel | C3×2- 1+4⋊C5 | 2- 1+4⋊C5 | C3×2- 1+4 | 2- 1+4 | C3 | C3 | C1 | C6 | C2 |
| # reps | 1 | 2 | 4 | 8 | 1 | 4 | 10 | 3 | 6 |
Matrix representation of C3×2- 1+4⋊C5 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 5 | 4 | 0 |
| 4 | 0 | 5 | 1 |
| 5 | 3 | 0 | 4 |
| 0 | 6 | 5 | 1 |
| 6 | 3 | 4 | 3 |
| 6 | 2 | 4 | 5 |
| 5 | 6 | 0 | 3 |
| 6 | 5 | 2 | 6 |
| 1 | 2 | 6 | 1 |
| 3 | 0 | 6 | 5 |
| 4 | 6 | 4 | 6 |
| 3 | 4 | 0 | 2 |
| 3 | 6 | 6 | 6 |
| 5 | 2 | 3 | 6 |
| 2 | 4 | 0 | 3 |
| 3 | 5 | 1 | 2 |
| 0 | 5 | 4 | 4 |
| 2 | 6 | 5 | 6 |
| 4 | 4 | 3 | 3 |
| 5 | 0 | 5 | 4 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,4,5,0,5,0,3,6,4,5,0,5,0,1,4,1],[6,6,5,6,3,2,6,5,4,4,0,2,3,5,3,6],[1,3,4,3,2,0,6,4,6,6,4,0,1,5,6,2],[3,5,2,3,6,2,4,5,6,3,0,1,6,6,3,2],[0,2,4,5,5,6,4,0,4,5,3,5,4,6,3,4] >;
C3×2- 1+4⋊C5 in GAP, Magma, Sage, TeX
C_3\times 2_-^{1+4}\rtimes C_5 % in TeX
G:=Group("C3xES-(2,2):C5"); // GroupNames label
G:=SmallGroup(480,1046);
// by ID
G=gap.SmallGroup(480,1046);
# by ID
G:=PCGroup([7,-3,-5,-2,2,2,2,-2,849,1270,521,248,1936,718,375,172,3162,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^4=c^2=f^5=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=b^2*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,e*d*e^-1=b^2*d,f*d*f^-1=b*c*d,f*e*f^-1=d*e>;
// generators/relations
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