direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4.Dic5, C60.180C23, C5⋊2C8⋊21D6, C5⋊9(S3×M4(2)), (S3×C20).7C4, C20.103(C4×S3), C60.100(C2×C4), (C4×S3).49D10, (C2×C20).312D6, (C2×C12).80D10, (C5×S3)⋊4M4(2), C15⋊16(C2×M4(2)), (C4×S3).1Dic5, D6.7(C2×Dic5), C60.7C4⋊10C2, C4.14(S3×Dic5), C15⋊3C8⋊26C22, D6.Dic5⋊13C2, (C2×C60).40C22, C12.10(C2×Dic5), C22.6(S3×Dic5), C6.3(C22×Dic5), (S3×C20).54C22, C20.177(C22×S3), C30.102(C22×C4), (C2×Dic3).5Dic5, (C10×Dic3).17C4, Dic3.5(C2×Dic5), C12.177(C22×D5), (C22×S3).3Dic5, (S3×C2×C4).1D5, (S3×C2×C20).1C2, C4.150(C2×S3×D5), (S3×C5⋊2C8)⋊12C2, C3⋊2(C2×C4.Dic5), C2.5(C2×S3×Dic5), (S3×C2×C10).10C4, C10.111(S3×C2×C4), (C2×C4).92(S3×D5), (C2×C30).99(C2×C4), (C2×C10).75(C4×S3), (C3×C4.Dic5)⋊5C2, (C2×C6).4(C2×Dic5), (S3×C10).34(C2×C4), (C3×C5⋊2C8)⋊21C22, (C5×Dic3).42(C2×C4), SmallGroup(480,363)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4.Dic5
G = < a,b,c,d,e | a3=b2=c4=1, d10=c2, e2=d5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d9 >
Subgroups: 412 in 136 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C5×S3, C30, C30, C2×M4(2), C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C22×C10, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C22×C20, S3×M4(2), C3×C5⋊2C8, C15⋊3C8, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C2×C4.Dic5, S3×C5⋊2C8, D6.Dic5, C3×C4.Dic5, C60.7C4, S3×C2×C20, S3×C4.Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, Dic5, D10, C4×S3, C22×S3, C2×M4(2), C2×Dic5, C22×D5, S3×C2×C4, S3×D5, C4.Dic5, C22×Dic5, S3×M4(2), S3×Dic5, C2×S3×D5, C2×C4.Dic5, C2×S3×Dic5, S3×C4.Dic5
►On 120 points
Generators in S120
(1 103 41)(2 104 42)(3 105 43)(4 106 44)(5 107 45)(6 108 46)(7 109 47)(8 110 48)(9 111 49)(10 112 50)(11 113 51)(12 114 52)(13 115 53)(14 116 54)(15 117 55)(16 118 56)(17 119 57)(18 120 58)(19 101 59)(20 102 60)(21 62 90)(22 63 91)(23 64 92)(24 65 93)(25 66 94)(26 67 95)(27 68 96)(28 69 97)(29 70 98)(30 71 99)(31 72 100)(32 73 81)(33 74 82)(34 75 83)(35 76 84)(36 77 85)(37 78 86)(38 79 87)(39 80 88)(40 61 89)
(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)
(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 74 6 79 11 64 16 69)(2 63 7 68 12 73 17 78)(3 72 8 77 13 62 18 67)(4 61 9 66 14 71 19 76)(5 70 10 75 15 80 20 65)(21 58 26 43 31 48 36 53)(22 47 27 52 32 57 37 42)(23 56 28 41 33 46 38 51)(24 45 29 50 34 55 39 60)(25 54 30 59 35 44 40 49)(81 119 86 104 91 109 96 114)(82 108 87 113 92 118 97 103)(83 117 88 102 93 107 98 112)(84 106 89 111 94 116 99 101)(85 115 90 120 95 105 100 110)
G:=sub<Sym(120)| (1,103,41)(2,104,42)(3,105,43)(4,106,44)(5,107,45)(6,108,46)(7,109,47)(8,110,48)(9,111,49)(10,112,50)(11,113,51)(12,114,52)(13,115,53)(14,116,54)(15,117,55)(16,118,56)(17,119,57)(18,120,58)(19,101,59)(20,102,60)(21,62,90)(22,63,91)(23,64,92)(24,65,93)(25,66,94)(26,67,95)(27,68,96)(28,69,97)(29,70,98)(30,71,99)(31,72,100)(32,73,81)(33,74,82)(34,75,83)(35,76,84)(36,77,85)(37,78,86)(38,79,87)(39,80,88)(40,61,89), (21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120), (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), (1,74,6,79,11,64,16,69)(2,63,7,68,12,73,17,78)(3,72,8,77,13,62,18,67)(4,61,9,66,14,71,19,76)(5,70,10,75,15,80,20,65)(21,58,26,43,31,48,36,53)(22,47,27,52,32,57,37,42)(23,56,28,41,33,46,38,51)(24,45,29,50,34,55,39,60)(25,54,30,59,35,44,40,49)(81,119,86,104,91,109,96,114)(82,108,87,113,92,118,97,103)(83,117,88,102,93,107,98,112)(84,106,89,111,94,116,99,101)(85,115,90,120,95,105,100,110)>;
G:=Group( (1,103,41)(2,104,42)(3,105,43)(4,106,44)(5,107,45)(6,108,46)(7,109,47)(8,110,48)(9,111,49)(10,112,50)(11,113,51)(12,114,52)(13,115,53)(14,116,54)(15,117,55)(16,118,56)(17,119,57)(18,120,58)(19,101,59)(20,102,60)(21,62,90)(22,63,91)(23,64,92)(24,65,93)(25,66,94)(26,67,95)(27,68,96)(28,69,97)(29,70,98)(30,71,99)(31,72,100)(32,73,81)(33,74,82)(34,75,83)(35,76,84)(36,77,85)(37,78,86)(38,79,87)(39,80,88)(40,61,89), (21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120), (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), (1,74,6,79,11,64,16,69)(2,63,7,68,12,73,17,78)(3,72,8,77,13,62,18,67)(4,61,9,66,14,71,19,76)(5,70,10,75,15,80,20,65)(21,58,26,43,31,48,36,53)(22,47,27,52,32,57,37,42)(23,56,28,41,33,46,38,51)(24,45,29,50,34,55,39,60)(25,54,30,59,35,44,40,49)(81,119,86,104,91,109,96,114)(82,108,87,113,92,118,97,103)(83,117,88,102,93,107,98,112)(84,106,89,111,94,116,99,101)(85,115,90,120,95,105,100,110) );
G=PermutationGroup([[(1,103,41),(2,104,42),(3,105,43),(4,106,44),(5,107,45),(6,108,46),(7,109,47),(8,110,48),(9,111,49),(10,112,50),(11,113,51),(12,114,52),(13,115,53),(14,116,54),(15,117,55),(16,118,56),(17,119,57),(18,120,58),(19,101,59),(20,102,60),(21,62,90),(22,63,91),(23,64,92),(24,65,93),(25,66,94),(26,67,95),(27,68,96),(28,69,97),(29,70,98),(30,71,99),(31,72,100),(32,73,81),(33,74,82),(34,75,83),(35,76,84),(36,77,85),(37,78,86),(38,79,87),(39,80,88),(40,61,89)], [(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,101),(60,102)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70),(81,96,91,86),(82,97,92,87),(83,98,93,88),(84,99,94,89),(85,100,95,90),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120)], [(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)], [(1,74,6,79,11,64,16,69),(2,63,7,68,12,73,17,78),(3,72,8,77,13,62,18,67),(4,61,9,66,14,71,19,76),(5,70,10,75,15,80,20,65),(21,58,26,43,31,48,36,53),(22,47,27,52,32,57,37,42),(23,56,28,41,33,46,38,51),(24,45,29,50,34,55,39,60),(25,54,30,59,35,44,40,49),(81,119,86,104,91,109,96,114),(82,108,87,113,92,118,97,103),(83,117,88,102,93,107,98,112),(84,106,89,111,94,116,99,101),(85,115,90,120,95,105,100,110)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | M4(2) | Dic5 | D10 | Dic5 | D10 | Dic5 | C4×S3 | C4×S3 | C4.Dic5 | S3×D5 | S3×M4(2) | S3×Dic5 | C2×S3×D5 | S3×Dic5 | S3×C4.Dic5 |
| kernel | S3×C4.Dic5 | S3×C5⋊2C8 | D6.Dic5 | C3×C4.Dic5 | C60.7C4 | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | C4.Dic5 | S3×C2×C4 | C5⋊2C8 | C2×C20 | C5×S3 | C4×S3 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C2×C10 | S3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of S3×C4.Dic5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 213 |
| 0 | 0 | 112 | 239 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 28 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,1,112,0,0,213,239],[240,0,0,0,0,240,0,0,0,0,240,0,0,0,28,1],[64,0,0,0,0,177,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,64,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
S3×C4.Dic5 in GAP, Magma, Sage, TeX
S_3\times C_4.{\rm Dic}_5 % in TeX
G:=Group("S3xC4.Dic5"); // GroupNames label
G:=SmallGroup(480,363);
// by ID
G=gap.SmallGroup(480,363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^10=c^2,e^2=d^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^9>;
// generators/relations