direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×C3⋊F5, C24⋊4F5, C120⋊4C4, C40⋊3Dic3, C30.8C42, C3⋊1(C8×F5), C15⋊4(C4×C8), C15⋊C8⋊7C4, C5⋊2(C8×Dic3), C6.1(C4×F5), (C8×D5).9S3, D5.2(S3×C8), C60.54(C2×C4), (C4×D5).93D6, C5⋊2C8⋊7Dic3, C12.56(C2×F5), (D5×C24).14C2, D10.12(C4×S3), C10.8(C4×Dic3), C60.C4.6C2, Dic5.17(C4×S3), C20.16(C2×Dic3), (D5×C12).121C22, C2.1(C4×C3⋊F5), (C2×C3⋊F5).4C4, (C4×C3⋊F5).6C2, C4.15(C2×C3⋊F5), (C3×C5⋊2C8)⋊13C4, (C3×D5).3(C2×C8), (C6×D5).39(C2×C4), (C3×Dic5).47(C2×C4), SmallGroup(480,296)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C8×C3⋊F5 |
Generators and relations for C8×C3⋊F5
G = < a,b,c,d | a8=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 364 in 88 conjugacy classes, 41 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C24, C24, C2×Dic3, C2×C12, C3×D5, C30, C4×C8, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C8×Dic3, C3×C5⋊2C8, C120, C15⋊C8, D5×C12, C2×C3⋊F5, C8×F5, D5×C24, C60.C4, C4×C3⋊F5, C8×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C4×S3, C2×Dic3, C4×C8, C2×F5, S3×C8, C4×Dic3, C3⋊F5, C4×F5, C8×Dic3, C2×C3⋊F5, C8×F5, C4×C3⋊F5, C8×C3⋊F5
►On 120 points
Generators in S120
(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 64 50)(2 57 51)(3 58 52)(4 59 53)(5 60 54)(6 61 55)(7 62 56)(8 63 49)(9 69 39)(10 70 40)(11 71 33)(12 72 34)(13 65 35)(14 66 36)(15 67 37)(16 68 38)(17 48 120)(18 41 113)(19 42 114)(20 43 115)(21 44 116)(22 45 117)(23 46 118)(24 47 119)(25 95 102)(26 96 103)(27 89 104)(28 90 97)(29 91 98)(30 92 99)(31 93 100)(32 94 101)(73 84 105)(74 85 106)(75 86 107)(76 87 108)(77 88 109)(78 81 110)(79 82 111)(80 83 112)
(1 73 113 65 99)(2 74 114 66 100)(3 75 115 67 101)(4 76 116 68 102)(5 77 117 69 103)(6 78 118 70 104)(7 79 119 71 97)(8 80 120 72 98)(9 96 54 109 45)(10 89 55 110 46)(11 90 56 111 47)(12 91 49 112 48)(13 92 50 105 41)(14 93 51 106 42)(15 94 52 107 43)(16 95 53 108 44)(17 34 29 63 83)(18 35 30 64 84)(19 36 31 57 85)(20 37 32 58 86)(21 38 25 59 87)(22 39 26 60 88)(23 40 27 61 81)(24 33 28 62 82)
(9 88 45 26)(10 81 46 27)(11 82 47 28)(12 83 48 29)(13 84 41 30)(14 85 42 31)(15 86 43 32)(16 87 44 25)(17 91 34 112)(18 92 35 105)(19 93 36 106)(20 94 37 107)(21 95 38 108)(22 96 39 109)(23 89 40 110)(24 90 33 111)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(65 73 113 99)(66 74 114 100)(67 75 115 101)(68 76 116 102)(69 77 117 103)(70 78 118 104)(71 79 119 97)(72 80 120 98)
G:=sub<Sym(120)| (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), (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (1,73,113,65,99)(2,74,114,66,100)(3,75,115,67,101)(4,76,116,68,102)(5,77,117,69,103)(6,78,118,70,104)(7,79,119,71,97)(8,80,120,72,98)(9,96,54,109,45)(10,89,55,110,46)(11,90,56,111,47)(12,91,49,112,48)(13,92,50,105,41)(14,93,51,106,42)(15,94,52,107,43)(16,95,53,108,44)(17,34,29,63,83)(18,35,30,64,84)(19,36,31,57,85)(20,37,32,58,86)(21,38,25,59,87)(22,39,26,60,88)(23,40,27,61,81)(24,33,28,62,82), (9,88,45,26)(10,81,46,27)(11,82,47,28)(12,83,48,29)(13,84,41,30)(14,85,42,31)(15,86,43,32)(16,87,44,25)(17,91,34,112)(18,92,35,105)(19,93,36,106)(20,94,37,107)(21,95,38,108)(22,96,39,109)(23,89,40,110)(24,90,33,111)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(65,73,113,99)(66,74,114,100)(67,75,115,101)(68,76,116,102)(69,77,117,103)(70,78,118,104)(71,79,119,97)(72,80,120,98)>;
G:=Group( (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), (1,64,50)(2,57,51)(3,58,52)(4,59,53)(5,60,54)(6,61,55)(7,62,56)(8,63,49)(9,69,39)(10,70,40)(11,71,33)(12,72,34)(13,65,35)(14,66,36)(15,67,37)(16,68,38)(17,48,120)(18,41,113)(19,42,114)(20,43,115)(21,44,116)(22,45,117)(23,46,118)(24,47,119)(25,95,102)(26,96,103)(27,89,104)(28,90,97)(29,91,98)(30,92,99)(31,93,100)(32,94,101)(73,84,105)(74,85,106)(75,86,107)(76,87,108)(77,88,109)(78,81,110)(79,82,111)(80,83,112), (1,73,113,65,99)(2,74,114,66,100)(3,75,115,67,101)(4,76,116,68,102)(5,77,117,69,103)(6,78,118,70,104)(7,79,119,71,97)(8,80,120,72,98)(9,96,54,109,45)(10,89,55,110,46)(11,90,56,111,47)(12,91,49,112,48)(13,92,50,105,41)(14,93,51,106,42)(15,94,52,107,43)(16,95,53,108,44)(17,34,29,63,83)(18,35,30,64,84)(19,36,31,57,85)(20,37,32,58,86)(21,38,25,59,87)(22,39,26,60,88)(23,40,27,61,81)(24,33,28,62,82), (9,88,45,26)(10,81,46,27)(11,82,47,28)(12,83,48,29)(13,84,41,30)(14,85,42,31)(15,86,43,32)(16,87,44,25)(17,91,34,112)(18,92,35,105)(19,93,36,106)(20,94,37,107)(21,95,38,108)(22,96,39,109)(23,89,40,110)(24,90,33,111)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(65,73,113,99)(66,74,114,100)(67,75,115,101)(68,76,116,102)(69,77,117,103)(70,78,118,104)(71,79,119,97)(72,80,120,98) );
G=PermutationGroup([[(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)], [(1,64,50),(2,57,51),(3,58,52),(4,59,53),(5,60,54),(6,61,55),(7,62,56),(8,63,49),(9,69,39),(10,70,40),(11,71,33),(12,72,34),(13,65,35),(14,66,36),(15,67,37),(16,68,38),(17,48,120),(18,41,113),(19,42,114),(20,43,115),(21,44,116),(22,45,117),(23,46,118),(24,47,119),(25,95,102),(26,96,103),(27,89,104),(28,90,97),(29,91,98),(30,92,99),(31,93,100),(32,94,101),(73,84,105),(74,85,106),(75,86,107),(76,87,108),(77,88,109),(78,81,110),(79,82,111),(80,83,112)], [(1,73,113,65,99),(2,74,114,66,100),(3,75,115,67,101),(4,76,116,68,102),(5,77,117,69,103),(6,78,118,70,104),(7,79,119,71,97),(8,80,120,72,98),(9,96,54,109,45),(10,89,55,110,46),(11,90,56,111,47),(12,91,49,112,48),(13,92,50,105,41),(14,93,51,106,42),(15,94,52,107,43),(16,95,53,108,44),(17,34,29,63,83),(18,35,30,64,84),(19,36,31,57,85),(20,37,32,58,86),(21,38,25,59,87),(22,39,26,60,88),(23,40,27,61,81),(24,33,28,62,82)], [(9,88,45,26),(10,81,46,27),(11,82,47,28),(12,83,48,29),(13,84,41,30),(14,85,42,31),(15,86,43,32),(16,87,44,25),(17,91,34,112),(18,92,35,105),(19,93,36,106),(20,94,37,107),(21,95,38,108),(22,96,39,109),(23,89,40,110),(24,90,33,111),(49,63),(50,64),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(65,73,113,99),(66,74,114,100),(67,75,115,101),(68,76,116,102),(69,77,117,103),(70,78,118,104),(71,79,119,97),(72,80,120,98)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 15 | ··· | 15 | 4 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | S3 | Dic3 | Dic3 | D6 | C4×S3 | C4×S3 | S3×C8 | F5 | C2×F5 | C3⋊F5 | C4×F5 | C2×C3⋊F5 | C8×F5 | C4×C3⋊F5 | C8×C3⋊F5 |
| kernel | C8×C3⋊F5 | D5×C24 | C60.C4 | C4×C3⋊F5 | C3×C5⋊2C8 | C120 | C15⋊C8 | C2×C3⋊F5 | C3⋊F5 | C8×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C8×C3⋊F5 ►in GL6(𝔽241)
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 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 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 126 | 0 | 12 | 12 |
| 0 | 0 | 229 | 114 | 229 | 0 |
| 0 | 0 | 0 | 229 | 114 | 229 |
| 0 | 0 | 12 | 12 | 0 | 126 |
| 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 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 64 | 64 | 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 |
G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,211,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,126,229,0,12,0,0,0,114,229,12,0,0,12,229,114,0,0,0,12,0,229,126],[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],[177,64,0,0,0,0,0,64,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
C8×C3⋊F5 in GAP, Magma, Sage, TeX
C_8\times C_3\rtimes F_5
% in TeX
G:=Group("C8xC3:F5"); // GroupNames label
G:=SmallGroup(480,296);
// by ID
G=gap.SmallGroup(480,296);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=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^-1,d*c*d^-1=c^3>;
// generators/relations