direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C8⋊F5, C24⋊7F5, C120⋊8C4, C40⋊4C12, C30.14C42, C5⋊C8⋊1C12, C8⋊3(C3×F5), C5⋊2C8⋊8C12, (C2×F5).C12, D5⋊C8.2C6, C15⋊5(C8⋊C4), (C6×F5).3C4, (C4×F5).2C6, C2.3(C12×F5), C6.14(C4×F5), C4.16(C6×F5), C10.2(C4×C12), C60.69(C2×C4), (C8×D5).10C6, (C12×F5).5C2, C12.69(C2×F5), D5.(C3×M4(2)), C20.16(C2×C12), (D5×C24).21C2, D10.6(C2×C12), Dic5.8(C2×C12), (C3×D5).4M4(2), (D5×C12).136C22, (C3×C5⋊C8)⋊5C4, C5⋊1(C3×C8⋊C4), (C3×C5⋊2C8)⋊18C4, (C3×D5⋊C8).5C2, (C6×D5).44(C2×C4), (C4×D5).33(C2×C6), (C3×Dic5).52(C2×C4), SmallGroup(480,272)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊F5
G = < a,b,c,d | a3=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >
Subgroups: 232 in 80 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C4×C12, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C3×C8⋊C4, C3×C5⋊2C8, C120, C3×C5⋊C8, D5×C12, C6×F5, C8⋊F5, D5×C24, C3×D5⋊C8, C12×F5, C3×C8⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, M4(2), F5, C2×C12, C8⋊C4, C2×F5, C4×C12, C3×M4(2), C3×F5, C4×F5, C3×C8⋊C4, C6×F5, C8⋊F5, C12×F5, C3×C8⋊F5
►On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 95 120)(18 96 113)(19 89 114)(20 90 115)(21 91 116)(22 92 117)(23 93 118)(24 94 119)(25 110 66)(26 111 67)(27 112 68)(28 105 69)(29 106 70)(30 107 71)(31 108 72)(32 109 65)(33 56 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 73)
(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)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)
(1 49 46 22 105)(2 50 47 23 106)(3 51 48 24 107)(4 52 41 17 108)(5 53 42 18 109)(6 54 43 19 110)(7 55 44 20 111)(8 56 45 21 112)(9 92 69 100 75)(10 93 70 101 76)(11 94 71 102 77)(12 95 72 103 78)(13 96 65 104 79)(14 89 66 97 80)(15 90 67 98 73)(16 91 68 99 74)(25 57 39 83 114)(26 58 40 84 115)(27 59 33 85 116)(28 60 34 86 117)(29 61 35 87 118)(30 62 36 88 119)(31 63 37 81 120)(32 64 38 82 113)
(2 6)(4 8)(9 69 92 75)(10 66 93 80)(11 71 94 77)(12 68 95 74)(13 65 96 79)(14 70 89 76)(15 67 90 73)(16 72 91 78)(17 56 41 112)(18 53 42 109)(19 50 43 106)(20 55 44 111)(21 52 45 108)(22 49 46 105)(23 54 47 110)(24 51 48 107)(25 118 39 87)(26 115 40 84)(27 120 33 81)(28 117 34 86)(29 114 35 83)(30 119 36 88)(31 116 37 85)(32 113 38 82)(57 61)(59 63)(97 101)(99 103)
G:=sub<Sym(120)| (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,95,120)(18,96,113)(19,89,114)(20,90,115)(21,91,116)(22,92,117)(23,93,118)(24,94,119)(25,110,66)(26,111,67)(27,112,68)(28,105,69)(29,106,70)(30,107,71)(31,108,72)(32,109,65)(33,56,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,73), (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)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120), (1,49,46,22,105)(2,50,47,23,106)(3,51,48,24,107)(4,52,41,17,108)(5,53,42,18,109)(6,54,43,19,110)(7,55,44,20,111)(8,56,45,21,112)(9,92,69,100,75)(10,93,70,101,76)(11,94,71,102,77)(12,95,72,103,78)(13,96,65,104,79)(14,89,66,97,80)(15,90,67,98,73)(16,91,68,99,74)(25,57,39,83,114)(26,58,40,84,115)(27,59,33,85,116)(28,60,34,86,117)(29,61,35,87,118)(30,62,36,88,119)(31,63,37,81,120)(32,64,38,82,113), (2,6)(4,8)(9,69,92,75)(10,66,93,80)(11,71,94,77)(12,68,95,74)(13,65,96,79)(14,70,89,76)(15,67,90,73)(16,72,91,78)(17,56,41,112)(18,53,42,109)(19,50,43,106)(20,55,44,111)(21,52,45,108)(22,49,46,105)(23,54,47,110)(24,51,48,107)(25,118,39,87)(26,115,40,84)(27,120,33,81)(28,117,34,86)(29,114,35,83)(30,119,36,88)(31,116,37,85)(32,113,38,82)(57,61)(59,63)(97,101)(99,103)>;
G:=Group( (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,95,120)(18,96,113)(19,89,114)(20,90,115)(21,91,116)(22,92,117)(23,93,118)(24,94,119)(25,110,66)(26,111,67)(27,112,68)(28,105,69)(29,106,70)(30,107,71)(31,108,72)(32,109,65)(33,56,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,73), (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)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120), (1,49,46,22,105)(2,50,47,23,106)(3,51,48,24,107)(4,52,41,17,108)(5,53,42,18,109)(6,54,43,19,110)(7,55,44,20,111)(8,56,45,21,112)(9,92,69,100,75)(10,93,70,101,76)(11,94,71,102,77)(12,95,72,103,78)(13,96,65,104,79)(14,89,66,97,80)(15,90,67,98,73)(16,91,68,99,74)(25,57,39,83,114)(26,58,40,84,115)(27,59,33,85,116)(28,60,34,86,117)(29,61,35,87,118)(30,62,36,88,119)(31,63,37,81,120)(32,64,38,82,113), (2,6)(4,8)(9,69,92,75)(10,66,93,80)(11,71,94,77)(12,68,95,74)(13,65,96,79)(14,70,89,76)(15,67,90,73)(16,72,91,78)(17,56,41,112)(18,53,42,109)(19,50,43,106)(20,55,44,111)(21,52,45,108)(22,49,46,105)(23,54,47,110)(24,51,48,107)(25,118,39,87)(26,115,40,84)(27,120,33,81)(28,117,34,86)(29,114,35,83)(30,119,36,88)(31,116,37,85)(32,113,38,82)(57,61)(59,63)(97,101)(99,103) );
G=PermutationGroup([[(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,46),(10,87,47),(11,88,48),(12,81,41),(13,82,42),(14,83,43),(15,84,44),(16,85,45),(17,95,120),(18,96,113),(19,89,114),(20,90,115),(21,91,116),(22,92,117),(23,93,118),(24,94,119),(25,110,66),(26,111,67),(27,112,68),(28,105,69),(29,106,70),(30,107,71),(31,108,72),(32,109,65),(33,56,74),(34,49,75),(35,50,76),(36,51,77),(37,52,78),(38,53,79),(39,54,80),(40,55,73)], [(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),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120)], [(1,49,46,22,105),(2,50,47,23,106),(3,51,48,24,107),(4,52,41,17,108),(5,53,42,18,109),(6,54,43,19,110),(7,55,44,20,111),(8,56,45,21,112),(9,92,69,100,75),(10,93,70,101,76),(11,94,71,102,77),(12,95,72,103,78),(13,96,65,104,79),(14,89,66,97,80),(15,90,67,98,73),(16,91,68,99,74),(25,57,39,83,114),(26,58,40,84,115),(27,59,33,85,116),(28,60,34,86,117),(29,61,35,87,118),(30,62,36,88,119),(31,63,37,81,120),(32,64,38,82,113)], [(2,6),(4,8),(9,69,92,75),(10,66,93,80),(11,71,94,77),(12,68,95,74),(13,65,96,79),(14,70,89,76),(15,67,90,73),(16,72,91,78),(17,56,41,112),(18,53,42,109),(19,50,43,106),(20,55,44,111),(21,52,45,108),(22,49,46,105),(23,54,47,110),(24,51,48,107),(25,118,39,87),(26,115,40,84),(27,120,33,81),(28,117,34,86),(29,114,35,83),(30,119,36,88),(31,116,37,85),(32,113,38,82),(57,61),(59,63),(97,101),(99,103)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | ··· | 8H | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | ··· | 24P | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | ··· | 10 | 4 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | C12 | M4(2) | C3×M4(2) | F5 | C2×F5 | C3×F5 | C4×F5 | C6×F5 | C8⋊F5 | C12×F5 | C3×C8⋊F5 |
| kernel | C3×C8⋊F5 | D5×C24 | C3×D5⋊C8 | C12×F5 | C8⋊F5 | C3×C5⋊2C8 | C120 | C3×C5⋊C8 | C6×F5 | C8×D5 | D5⋊C8 | C4×F5 | C5⋊2C8 | C40 | C5⋊C8 | C2×F5 | C3×D5 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C8⋊F5 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 240 | 128 | 0 | 0 | 0 | 0 |
| 152 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 0 | 177 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 1 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 1 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[240,152,0,0,0,0,128,1,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[64,1,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,240,240,240,240,0,0,0,1,0,0] >;
C3×C8⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes F_5
% in TeX
G:=Group("C3xC8:F5"); // GroupNames label
G:=SmallGroup(480,272);
// by ID
G=gap.SmallGroup(480,272);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations