direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×S3×M4(2), C40⋊28D6, C120⋊35C22, C60.286C23, C8⋊6(S3×C10), (S3×C8)⋊7C10, C24⋊7(C2×C10), (S3×C40)⋊16C2, C8⋊S3⋊5C10, (C4×S3).1C20, C4.15(S3×C20), D6.9(C2×C20), C3⋊2(C10×M4(2)), (S3×C20).12C4, C20.117(C4×S3), C60.182(C2×C4), C12.12(C2×C20), (C2×C20).356D6, C15⋊28(C2×M4(2)), C4.Dic3⋊5C10, C22.7(S3×C20), (C3×M4(2))⋊5C10, (C22×S3).4C20, C6.15(C22×C20), Dic3.8(C2×C20), (C2×Dic3).6C20, (C15×M4(2))⋊13C2, (S3×C20).65C22, C20.244(C22×S3), C12.38(C22×C10), (C2×C60).353C22, C30.206(C22×C4), (C10×Dic3).23C4, C3⋊C8⋊11(C2×C10), (S3×C2×C4).4C10, C4.38(S3×C2×C10), C2.16(S3×C2×C20), (C5×C3⋊C8)⋊44C22, (S3×C2×C20).15C2, (S3×C2×C10).14C4, C10.142(S3×C2×C4), (C2×C6).5(C2×C20), (C5×C8⋊S3)⋊13C2, (C2×C10).66(C4×S3), (C2×C4).45(S3×C10), (S3×C10).45(C2×C4), (C4×S3).16(C2×C10), (C2×C12).26(C2×C10), (C2×C30).129(C2×C4), (C5×C4.Dic3)⋊17C2, (C5×Dic3).50(C2×C4), SmallGroup(480,785)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×S3×M4(2)
G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >
Subgroups: 260 in 136 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), 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), C40, C40, 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×C40, C5×M4(2), C5×M4(2), C22×C20, S3×M4(2), C5×C3⋊C8, C120, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C10×M4(2), S3×C40, C5×C8⋊S3, C5×C4.Dic3, C15×M4(2), S3×C2×C20, C5×S3×M4(2)
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, M4(2), C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C2×M4(2), C2×C20, C22×C10, S3×C2×C4, S3×C10, C5×M4(2), C22×C20, S3×M4(2), S3×C20, S3×C2×C10, C10×M4(2), S3×C2×C20, C5×S3×M4(2)
►On 120 points
Generators in S120
(1 58 41 51 24)(2 59 42 52 17)(3 60 43 53 18)(4 61 44 54 19)(5 62 45 55 20)(6 63 46 56 21)(7 64 47 49 22)(8 57 48 50 23)(9 85 110 75 36)(10 86 111 76 37)(11 87 112 77 38)(12 88 105 78 39)(13 81 106 79 40)(14 82 107 80 33)(15 83 108 73 34)(16 84 109 74 35)(25 101 72 92 117)(26 102 65 93 118)(27 103 66 94 119)(28 104 67 95 120)(29 97 68 96 113)(30 98 69 89 114)(31 99 70 90 115)(32 100 71 91 116)
(1 75 99)(2 76 100)(3 77 101)(4 78 102)(5 79 103)(6 80 104)(7 73 97)(8 74 98)(9 90 41)(10 91 42)(11 92 43)(12 93 44)(13 94 45)(14 95 46)(15 96 47)(16 89 48)(17 111 32)(18 112 25)(19 105 26)(20 106 27)(21 107 28)(22 108 29)(23 109 30)(24 110 31)(33 67 63)(34 68 64)(35 69 57)(36 70 58)(37 71 59)(38 72 60)(39 65 61)(40 66 62)(49 83 113)(50 84 114)(51 85 115)(52 86 116)(53 87 117)(54 88 118)(55 81 119)(56 82 120)
(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 89)(25 112)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 119)(82 120)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)
G:=sub<Sym(120)| (1,58,41,51,24)(2,59,42,52,17)(3,60,43,53,18)(4,61,44,54,19)(5,62,45,55,20)(6,63,46,56,21)(7,64,47,49,22)(8,57,48,50,23)(9,85,110,75,36)(10,86,111,76,37)(11,87,112,77,38)(12,88,105,78,39)(13,81,106,79,40)(14,82,107,80,33)(15,83,108,73,34)(16,84,109,74,35)(25,101,72,92,117)(26,102,65,93,118)(27,103,66,94,119)(28,104,67,95,120)(29,97,68,96,113)(30,98,69,89,114)(31,99,70,90,115)(32,100,71,91,116), (1,75,99)(2,76,100)(3,77,101)(4,78,102)(5,79,103)(6,80,104)(7,73,97)(8,74,98)(9,90,41)(10,91,42)(11,92,43)(12,93,44)(13,94,45)(14,95,46)(15,96,47)(16,89,48)(17,111,32)(18,112,25)(19,105,26)(20,106,27)(21,107,28)(22,108,29)(23,109,30)(24,110,31)(33,67,63)(34,68,64)(35,69,57)(36,70,58)(37,71,59)(38,72,60)(39,65,61)(40,66,62)(49,83,113)(50,84,114)(51,85,115)(52,86,116)(53,87,117)(54,88,118)(55,81,119)(56,82,120), (9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,89)(25,112)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120)>;
G:=Group( (1,58,41,51,24)(2,59,42,52,17)(3,60,43,53,18)(4,61,44,54,19)(5,62,45,55,20)(6,63,46,56,21)(7,64,47,49,22)(8,57,48,50,23)(9,85,110,75,36)(10,86,111,76,37)(11,87,112,77,38)(12,88,105,78,39)(13,81,106,79,40)(14,82,107,80,33)(15,83,108,73,34)(16,84,109,74,35)(25,101,72,92,117)(26,102,65,93,118)(27,103,66,94,119)(28,104,67,95,120)(29,97,68,96,113)(30,98,69,89,114)(31,99,70,90,115)(32,100,71,91,116), (1,75,99)(2,76,100)(3,77,101)(4,78,102)(5,79,103)(6,80,104)(7,73,97)(8,74,98)(9,90,41)(10,91,42)(11,92,43)(12,93,44)(13,94,45)(14,95,46)(15,96,47)(16,89,48)(17,111,32)(18,112,25)(19,105,26)(20,106,27)(21,107,28)(22,108,29)(23,109,30)(24,110,31)(33,67,63)(34,68,64)(35,69,57)(36,70,58)(37,71,59)(38,72,60)(39,65,61)(40,66,62)(49,83,113)(50,84,114)(51,85,115)(52,86,116)(53,87,117)(54,88,118)(55,81,119)(56,82,120), (9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,89)(25,112)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120) );
G=PermutationGroup([[(1,58,41,51,24),(2,59,42,52,17),(3,60,43,53,18),(4,61,44,54,19),(5,62,45,55,20),(6,63,46,56,21),(7,64,47,49,22),(8,57,48,50,23),(9,85,110,75,36),(10,86,111,76,37),(11,87,112,77,38),(12,88,105,78,39),(13,81,106,79,40),(14,82,107,80,33),(15,83,108,73,34),(16,84,109,74,35),(25,101,72,92,117),(26,102,65,93,118),(27,103,66,94,119),(28,104,67,95,120),(29,97,68,96,113),(30,98,69,89,114),(31,99,70,90,115),(32,100,71,91,116)], [(1,75,99),(2,76,100),(3,77,101),(4,78,102),(5,79,103),(6,80,104),(7,73,97),(8,74,98),(9,90,41),(10,91,42),(11,92,43),(12,93,44),(13,94,45),(14,95,46),(15,96,47),(16,89,48),(17,111,32),(18,112,25),(19,105,26),(20,106,27),(21,107,28),(22,108,29),(23,109,30),(24,110,31),(33,67,63),(34,68,64),(35,69,57),(36,70,58),(37,71,59),(38,72,60),(39,65,61),(40,66,62),(49,83,113),(50,84,114),(51,85,115),(52,86,116),(53,87,117),(54,88,118),(55,81,119),(56,82,120)], [(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,89),(25,112),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,65),(40,66),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,119),(82,120),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(65,69),(67,71),(74,78),(76,80),(82,86),(84,88),(89,93),(91,95),(98,102),(100,104),(105,109),(107,111),(114,118),(116,120)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 10Q | 10R | 10S | 10T | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 20U | 20V | 20W | 20X | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 1 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C5×S3 | S3×C10 | S3×C10 | C5×M4(2) | S3×C20 | S3×C20 | S3×M4(2) | C5×S3×M4(2) |
| kernel | C5×S3×M4(2) | S3×C40 | C5×C8⋊S3 | C5×C4.Dic3 | C15×M4(2) | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | S3×M4(2) | S3×C8 | C8⋊S3 | C4.Dic3 | C3×M4(2) | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | C5×M4(2) | C40 | C2×C20 | C5×S3 | C20 | C2×C10 | M4(2) | C8 | C2×C4 | S3 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 16 | 8 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 4 | 16 | 8 | 8 | 2 | 8 |
Matrix representation of C5×S3×M4(2) ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 239 |
| 0 | 0 | 126 | 195 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 240 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,91,0,0,0,0,91],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,126,0,0,239,195],[1,0,0,0,0,1,0,0,0,0,1,46,0,0,0,240] >;
C5×S3×M4(2) in GAP, Magma, Sage, TeX
C_5\times S_3\times M_4(2)
% in TeX
G:=Group("C5xS3xM4(2)"); // GroupNames label
G:=SmallGroup(480,785);
// by ID
G=gap.SmallGroup(480,785);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,891,226,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations