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
| C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — C3×2- (1+4)⋊C5 |
| 2- (1+4) — C3×2- (1+4)⋊C5 |
Subgroups: 222 in 40 conjugacy classes, 8 normal (all characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C2×C4 [×3], D4 [×2], Q8 [×2], C10, C12 [×2], C2×C6, C15, C2×Q8, C4○D4 [×2], C2×C12 [×3], C3×D4 [×2], C3×Q8 [×2], C30, 2- (1+4), C6×Q8, C3×C4○D4 [×2], C3×2- (1+4), 2- (1+4)⋊C5, C3×2- (1+4)⋊C5
Quotients:
C1, C3, C5, C15, C24⋊C5, 2- (1+4)⋊C5, C3×C24⋊C5, C3×2- (1+4)⋊C5
Generators and relations
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 >
►On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 43 69)(8 44 70)(9 45 71)(10 46 67)(11 42 68)(12 38 73)(13 39 74)(14 40 75)(15 41 76)(16 37 72)(17 84 79)(18 85 80)(19 86 81)(20 82 77)(21 83 78)(22 35 55)(23 36 56)(24 32 52)(25 33 53)(26 34 54)(27 64 61)(28 65 57)(29 66 58)(30 62 59)(31 63 60)(47 87 96)(48 88 92)(49 89 93)(50 90 94)(51 91 95)
(1 81 4 9)(2 45 5 19)(3 86 6 71)(7 74 79 25)(8 48 80 28)(10 29 77 49)(11 76 78 22)(12 36 32 16)(13 17 33 43)(14 91 34 63)(15 21 35 42)(18 65 44 88)(20 89 46 66)(23 24 72 73)(26 31 75 51)(27 50 47 30)(37 38 56 52)(39 84 53 69)(40 95 54 60)(41 83 55 68)(57 70 92 85)(58 82 93 67)(59 61 94 96)(62 64 90 87)
(1 24)(2 12)(3 52)(4 73)(5 32)(6 38)(7 49)(8 78)(9 72)(10 25)(11 80)(13 20)(14 62)(15 88)(16 45)(17 66)(18 42)(19 36)(21 44)(22 28)(23 81)(26 50)(27 31)(29 79)(30 75)(33 46)(34 90)(35 65)(37 71)(39 82)(40 59)(41 92)(43 89)(47 51)(48 76)(53 67)(54 94)(55 57)(56 86)(58 84)(60 61)(63 64)(68 85)(69 93)(70 83)(74 77)(87 91)(95 96)
(1 51 4 31)(2 63 5 91)(3 95 6 60)(7 78 79 11)(8 29 80 49)(9 75 81 26)(10 48 77 28)(12 64 32 87)(13 35 33 15)(14 19 34 45)(16 62 36 90)(17 42 43 21)(18 89 44 66)(20 65 46 88)(22 25 76 74)(23 50 72 30)(24 47 73 27)(37 59 56 94)(38 61 52 96)(39 55 53 41)(40 86 54 71)(57 67 92 82)(58 85 93 70)(68 69 83 84)
(1 79 4 7)(2 43 5 17)(3 84 6 69)(8 27 80 47)(9 74 81 25)(10 72 77 23)(11 51 78 31)(12 89 32 66)(13 19 33 45)(14 15 34 35)(16 20 36 46)(18 87 44 64)(21 63 42 91)(22 75 76 26)(24 29 73 49)(28 30 48 50)(37 82 56 67)(38 93 52 58)(39 86 53 71)(40 41 54 55)(57 59 92 94)(60 68 95 83)(61 85 96 70)(62 88 90 65)
(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)| (1,5,3)(2,6,4)(7,43,69)(8,44,70)(9,45,71)(10,46,67)(11,42,68)(12,38,73)(13,39,74)(14,40,75)(15,41,76)(16,37,72)(17,84,79)(18,85,80)(19,86,81)(20,82,77)(21,83,78)(22,35,55)(23,36,56)(24,32,52)(25,33,53)(26,34,54)(27,64,61)(28,65,57)(29,66,58)(30,62,59)(31,63,60)(47,87,96)(48,88,92)(49,89,93)(50,90,94)(51,91,95), (1,81,4,9)(2,45,5,19)(3,86,6,71)(7,74,79,25)(8,48,80,28)(10,29,77,49)(11,76,78,22)(12,36,32,16)(13,17,33,43)(14,91,34,63)(15,21,35,42)(18,65,44,88)(20,89,46,66)(23,24,72,73)(26,31,75,51)(27,50,47,30)(37,38,56,52)(39,84,53,69)(40,95,54,60)(41,83,55,68)(57,70,92,85)(58,82,93,67)(59,61,94,96)(62,64,90,87), (1,24)(2,12)(3,52)(4,73)(5,32)(6,38)(7,49)(8,78)(9,72)(10,25)(11,80)(13,20)(14,62)(15,88)(16,45)(17,66)(18,42)(19,36)(21,44)(22,28)(23,81)(26,50)(27,31)(29,79)(30,75)(33,46)(34,90)(35,65)(37,71)(39,82)(40,59)(41,92)(43,89)(47,51)(48,76)(53,67)(54,94)(55,57)(56,86)(58,84)(60,61)(63,64)(68,85)(69,93)(70,83)(74,77)(87,91)(95,96), (1,51,4,31)(2,63,5,91)(3,95,6,60)(7,78,79,11)(8,29,80,49)(9,75,81,26)(10,48,77,28)(12,64,32,87)(13,35,33,15)(14,19,34,45)(16,62,36,90)(17,42,43,21)(18,89,44,66)(20,65,46,88)(22,25,76,74)(23,50,72,30)(24,47,73,27)(37,59,56,94)(38,61,52,96)(39,55,53,41)(40,86,54,71)(57,67,92,82)(58,85,93,70)(68,69,83,84), (1,79,4,7)(2,43,5,17)(3,84,6,69)(8,27,80,47)(9,74,81,25)(10,72,77,23)(11,51,78,31)(12,89,32,66)(13,19,33,45)(14,15,34,35)(16,20,36,46)(18,87,44,64)(21,63,42,91)(22,75,76,26)(24,29,73,49)(28,30,48,50)(37,82,56,67)(38,93,52,58)(39,86,53,71)(40,41,54,55)(57,59,92,94)(60,68,95,83)(61,85,96,70)(62,88,90,65), (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( (1,5,3)(2,6,4)(7,43,69)(8,44,70)(9,45,71)(10,46,67)(11,42,68)(12,38,73)(13,39,74)(14,40,75)(15,41,76)(16,37,72)(17,84,79)(18,85,80)(19,86,81)(20,82,77)(21,83,78)(22,35,55)(23,36,56)(24,32,52)(25,33,53)(26,34,54)(27,64,61)(28,65,57)(29,66,58)(30,62,59)(31,63,60)(47,87,96)(48,88,92)(49,89,93)(50,90,94)(51,91,95), (1,81,4,9)(2,45,5,19)(3,86,6,71)(7,74,79,25)(8,48,80,28)(10,29,77,49)(11,76,78,22)(12,36,32,16)(13,17,33,43)(14,91,34,63)(15,21,35,42)(18,65,44,88)(20,89,46,66)(23,24,72,73)(26,31,75,51)(27,50,47,30)(37,38,56,52)(39,84,53,69)(40,95,54,60)(41,83,55,68)(57,70,92,85)(58,82,93,67)(59,61,94,96)(62,64,90,87), (1,24)(2,12)(3,52)(4,73)(5,32)(6,38)(7,49)(8,78)(9,72)(10,25)(11,80)(13,20)(14,62)(15,88)(16,45)(17,66)(18,42)(19,36)(21,44)(22,28)(23,81)(26,50)(27,31)(29,79)(30,75)(33,46)(34,90)(35,65)(37,71)(39,82)(40,59)(41,92)(43,89)(47,51)(48,76)(53,67)(54,94)(55,57)(56,86)(58,84)(60,61)(63,64)(68,85)(69,93)(70,83)(74,77)(87,91)(95,96), (1,51,4,31)(2,63,5,91)(3,95,6,60)(7,78,79,11)(8,29,80,49)(9,75,81,26)(10,48,77,28)(12,64,32,87)(13,35,33,15)(14,19,34,45)(16,62,36,90)(17,42,43,21)(18,89,44,66)(20,65,46,88)(22,25,76,74)(23,50,72,30)(24,47,73,27)(37,59,56,94)(38,61,52,96)(39,55,53,41)(40,86,54,71)(57,67,92,82)(58,85,93,70)(68,69,83,84), (1,79,4,7)(2,43,5,17)(3,84,6,69)(8,27,80,47)(9,74,81,25)(10,72,77,23)(11,51,78,31)(12,89,32,66)(13,19,33,45)(14,15,34,35)(16,20,36,46)(18,87,44,64)(21,63,42,91)(22,75,76,26)(24,29,73,49)(28,30,48,50)(37,82,56,67)(38,93,52,58)(39,86,53,71)(40,41,54,55)(57,59,92,94)(60,68,95,83)(61,85,96,70)(62,88,90,65), (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([(1,5,3),(2,6,4),(7,43,69),(8,44,70),(9,45,71),(10,46,67),(11,42,68),(12,38,73),(13,39,74),(14,40,75),(15,41,76),(16,37,72),(17,84,79),(18,85,80),(19,86,81),(20,82,77),(21,83,78),(22,35,55),(23,36,56),(24,32,52),(25,33,53),(26,34,54),(27,64,61),(28,65,57),(29,66,58),(30,62,59),(31,63,60),(47,87,96),(48,88,92),(49,89,93),(50,90,94),(51,91,95)], [(1,81,4,9),(2,45,5,19),(3,86,6,71),(7,74,79,25),(8,48,80,28),(10,29,77,49),(11,76,78,22),(12,36,32,16),(13,17,33,43),(14,91,34,63),(15,21,35,42),(18,65,44,88),(20,89,46,66),(23,24,72,73),(26,31,75,51),(27,50,47,30),(37,38,56,52),(39,84,53,69),(40,95,54,60),(41,83,55,68),(57,70,92,85),(58,82,93,67),(59,61,94,96),(62,64,90,87)], [(1,24),(2,12),(3,52),(4,73),(5,32),(6,38),(7,49),(8,78),(9,72),(10,25),(11,80),(13,20),(14,62),(15,88),(16,45),(17,66),(18,42),(19,36),(21,44),(22,28),(23,81),(26,50),(27,31),(29,79),(30,75),(33,46),(34,90),(35,65),(37,71),(39,82),(40,59),(41,92),(43,89),(47,51),(48,76),(53,67),(54,94),(55,57),(56,86),(58,84),(60,61),(63,64),(68,85),(69,93),(70,83),(74,77),(87,91),(95,96)], [(1,51,4,31),(2,63,5,91),(3,95,6,60),(7,78,79,11),(8,29,80,49),(9,75,81,26),(10,48,77,28),(12,64,32,87),(13,35,33,15),(14,19,34,45),(16,62,36,90),(17,42,43,21),(18,89,44,66),(20,65,46,88),(22,25,76,74),(23,50,72,30),(24,47,73,27),(37,59,56,94),(38,61,52,96),(39,55,53,41),(40,86,54,71),(57,67,92,82),(58,85,93,70),(68,69,83,84)], [(1,79,4,7),(2,43,5,17),(3,84,6,69),(8,27,80,47),(9,74,81,25),(10,72,77,23),(11,51,78,31),(12,89,32,66),(13,19,33,45),(14,15,34,35),(16,20,36,46),(18,87,44,64),(21,63,42,91),(22,75,76,26),(24,29,73,49),(28,30,48,50),(37,82,56,67),(38,93,52,58),(39,86,53,71),(40,41,54,55),(57,59,92,94),(60,68,95,83),(61,85,96,70),(62,88,90,65)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀