direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×C3⋊C8, C30.1C42, (C3×F5)⋊C8, C3⋊3(C8×F5), C15⋊1(C4×C8), C15⋊3C8⋊3C4, C15⋊C8⋊1C4, C6.6(C4×F5), (C6×F5).1C4, (C4×F5).3S3, C4.22(S3×F5), C20.22(C4×S3), C60.22(C2×C4), (C4×D5).68D6, (C12×F5).2C2, C12.29(C2×F5), C2.1(Dic3×F5), Dic5.9(C4×S3), (C2×F5).2Dic3, C10.1(C4×Dic3), C60.C4.2C2, D10.5(C2×Dic3), (D5×C12).60C22, C5⋊1(C4×C3⋊C8), D5.(C2×C3⋊C8), (C5×C3⋊C8)⋊3C4, (D5×C3⋊C8).6C2, (C6×D5).9(C2×C4), (C3×D5).1(C2×C8), (C3×Dic5).13(C2×C4), SmallGroup(480,223)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — F5×C3⋊C8 |
Generators and relations for F5×C3⋊C8
G = < a,b,c,d | a5=b4=c3=d8=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 308 in 88 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C2×C12, C3×D5, C30, C4×C8, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C4×C3⋊C8, C5×C3⋊C8, C15⋊3C8, C15⋊C8, D5×C12, C6×F5, C8×F5, D5×C3⋊C8, C12×F5, C60.C4, F5×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C3⋊C8, C4×S3, C2×Dic3, C4×C8, C2×F5, C2×C3⋊C8, C4×Dic3, C4×F5, C4×C3⋊C8, S3×F5, C8×F5, Dic3×F5, F5×C3⋊C8
►On 120 points
Generators in S120
(1 99 88 113 28)(2 100 81 114 29)(3 101 82 115 30)(4 102 83 116 31)(5 103 84 117 32)(6 104 85 118 25)(7 97 86 119 26)(8 98 87 120 27)(9 43 72 39 22)(10 44 65 40 23)(11 45 66 33 24)(12 46 67 34 17)(13 47 68 35 18)(14 48 69 36 19)(15 41 70 37 20)(16 42 71 38 21)(49 61 95 78 112)(50 62 96 79 105)(51 63 89 80 106)(52 64 90 73 107)(53 57 91 74 108)(54 58 92 75 109)(55 59 93 76 110)(56 60 94 77 111)
(17 34 46 67)(18 35 47 68)(19 36 48 69)(20 37 41 70)(21 38 42 71)(22 39 43 72)(23 40 44 65)(24 33 45 66)(25 118 104 85)(26 119 97 86)(27 120 98 87)(28 113 99 88)(29 114 100 81)(30 115 101 82)(31 116 102 83)(32 117 103 84)(57 91 108 74)(58 92 109 75)(59 93 110 76)(60 94 111 77)(61 95 112 78)(62 96 105 79)(63 89 106 80)(64 90 107 73)
(1 11 50)(2 51 12)(3 13 52)(4 53 14)(5 15 54)(6 55 16)(7 9 56)(8 49 10)(17 29 106)(18 107 30)(19 31 108)(20 109 32)(21 25 110)(22 111 26)(23 27 112)(24 105 28)(33 79 113)(34 114 80)(35 73 115)(36 116 74)(37 75 117)(38 118 76)(39 77 119)(40 120 78)(41 58 103)(42 104 59)(43 60 97)(44 98 61)(45 62 99)(46 100 63)(47 64 101)(48 102 57)(65 87 95)(66 96 88)(67 81 89)(68 90 82)(69 83 91)(70 92 84)(71 85 93)(72 94 86)
(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)
G:=sub<Sym(120)| (1,99,88,113,28)(2,100,81,114,29)(3,101,82,115,30)(4,102,83,116,31)(5,103,84,117,32)(6,104,85,118,25)(7,97,86,119,26)(8,98,87,120,27)(9,43,72,39,22)(10,44,65,40,23)(11,45,66,33,24)(12,46,67,34,17)(13,47,68,35,18)(14,48,69,36,19)(15,41,70,37,20)(16,42,71,38,21)(49,61,95,78,112)(50,62,96,79,105)(51,63,89,80,106)(52,64,90,73,107)(53,57,91,74,108)(54,58,92,75,109)(55,59,93,76,110)(56,60,94,77,111), (17,34,46,67)(18,35,47,68)(19,36,48,69)(20,37,41,70)(21,38,42,71)(22,39,43,72)(23,40,44,65)(24,33,45,66)(25,118,104,85)(26,119,97,86)(27,120,98,87)(28,113,99,88)(29,114,100,81)(30,115,101,82)(31,116,102,83)(32,117,103,84)(57,91,108,74)(58,92,109,75)(59,93,110,76)(60,94,111,77)(61,95,112,78)(62,96,105,79)(63,89,106,80)(64,90,107,73), (1,11,50)(2,51,12)(3,13,52)(4,53,14)(5,15,54)(6,55,16)(7,9,56)(8,49,10)(17,29,106)(18,107,30)(19,31,108)(20,109,32)(21,25,110)(22,111,26)(23,27,112)(24,105,28)(33,79,113)(34,114,80)(35,73,115)(36,116,74)(37,75,117)(38,118,76)(39,77,119)(40,120,78)(41,58,103)(42,104,59)(43,60,97)(44,98,61)(45,62,99)(46,100,63)(47,64,101)(48,102,57)(65,87,95)(66,96,88)(67,81,89)(68,90,82)(69,83,91)(70,92,84)(71,85,93)(72,94,86), (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)>;
G:=Group( (1,99,88,113,28)(2,100,81,114,29)(3,101,82,115,30)(4,102,83,116,31)(5,103,84,117,32)(6,104,85,118,25)(7,97,86,119,26)(8,98,87,120,27)(9,43,72,39,22)(10,44,65,40,23)(11,45,66,33,24)(12,46,67,34,17)(13,47,68,35,18)(14,48,69,36,19)(15,41,70,37,20)(16,42,71,38,21)(49,61,95,78,112)(50,62,96,79,105)(51,63,89,80,106)(52,64,90,73,107)(53,57,91,74,108)(54,58,92,75,109)(55,59,93,76,110)(56,60,94,77,111), (17,34,46,67)(18,35,47,68)(19,36,48,69)(20,37,41,70)(21,38,42,71)(22,39,43,72)(23,40,44,65)(24,33,45,66)(25,118,104,85)(26,119,97,86)(27,120,98,87)(28,113,99,88)(29,114,100,81)(30,115,101,82)(31,116,102,83)(32,117,103,84)(57,91,108,74)(58,92,109,75)(59,93,110,76)(60,94,111,77)(61,95,112,78)(62,96,105,79)(63,89,106,80)(64,90,107,73), (1,11,50)(2,51,12)(3,13,52)(4,53,14)(5,15,54)(6,55,16)(7,9,56)(8,49,10)(17,29,106)(18,107,30)(19,31,108)(20,109,32)(21,25,110)(22,111,26)(23,27,112)(24,105,28)(33,79,113)(34,114,80)(35,73,115)(36,116,74)(37,75,117)(38,118,76)(39,77,119)(40,120,78)(41,58,103)(42,104,59)(43,60,97)(44,98,61)(45,62,99)(46,100,63)(47,64,101)(48,102,57)(65,87,95)(66,96,88)(67,81,89)(68,90,82)(69,83,91)(70,92,84)(71,85,93)(72,94,86), (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) );
G=PermutationGroup([[(1,99,88,113,28),(2,100,81,114,29),(3,101,82,115,30),(4,102,83,116,31),(5,103,84,117,32),(6,104,85,118,25),(7,97,86,119,26),(8,98,87,120,27),(9,43,72,39,22),(10,44,65,40,23),(11,45,66,33,24),(12,46,67,34,17),(13,47,68,35,18),(14,48,69,36,19),(15,41,70,37,20),(16,42,71,38,21),(49,61,95,78,112),(50,62,96,79,105),(51,63,89,80,106),(52,64,90,73,107),(53,57,91,74,108),(54,58,92,75,109),(55,59,93,76,110),(56,60,94,77,111)], [(17,34,46,67),(18,35,47,68),(19,36,48,69),(20,37,41,70),(21,38,42,71),(22,39,43,72),(23,40,44,65),(24,33,45,66),(25,118,104,85),(26,119,97,86),(27,120,98,87),(28,113,99,88),(29,114,100,81),(30,115,101,82),(31,116,102,83),(32,117,103,84),(57,91,108,74),(58,92,109,75),(59,93,110,76),(60,94,111,77),(61,95,112,78),(62,96,105,79),(63,89,106,80),(64,90,107,73)], [(1,11,50),(2,51,12),(3,13,52),(4,53,14),(5,15,54),(6,55,16),(7,9,56),(8,49,10),(17,29,106),(18,107,30),(19,31,108),(20,109,32),(21,25,110),(22,111,26),(23,27,112),(24,105,28),(33,79,113),(34,114,80),(35,73,115),(36,116,74),(37,75,117),(38,118,76),(39,77,119),(40,120,78),(41,58,103),(42,104,59),(43,60,97),(44,98,61),(45,62,99),(46,100,63),(47,64,101),(48,102,57),(65,87,95),(66,96,88),(67,81,89),(68,90,82),(69,83,91),(70,92,84),(71,85,93),(72,94,86)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | ··· | 4L | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8P | 10 | 12A | 12B | 12C | ··· | 12L | 15 | 20A | 20B | 30 | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | ··· | 5 | 4 | 2 | 10 | 10 | 3 | 3 | 3 | 3 | 15 | ··· | 15 | 4 | 2 | 2 | 10 | ··· | 10 | 8 | 4 | 4 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | S3 | D6 | Dic3 | C4×S3 | C4×S3 | C3⋊C8 | F5 | C2×F5 | C4×F5 | C8×F5 | S3×F5 | Dic3×F5 | F5×C3⋊C8 |
| kernel | F5×C3⋊C8 | D5×C3⋊C8 | C12×F5 | C60.C4 | C5×C3⋊C8 | C15⋊3C8 | C15⋊C8 | C6×F5 | C3×F5 | C4×F5 | C4×D5 | C2×F5 | Dic5 | C20 | F5 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of F5×C3⋊C8 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 186 | 78 | 0 | 0 | 0 | 0 |
| 23 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[186,23,0,0,0,0,78,55,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
F5×C3⋊C8 in GAP, Magma, Sage, TeX
F_5\times C_3\rtimes C_8
% in TeX
G:=Group("F5xC3:C8"); // GroupNames label
G:=SmallGroup(480,223);
// by ID
G=gap.SmallGroup(480,223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^3=d^8=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations