direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D5×M4(2), C24⋊28D10, C120⋊33C22, C60.277C23, C8⋊6(C6×D5), C40⋊6(C2×C6), (C8×D5)⋊7C6, C8⋊D5⋊5C6, C5⋊5(C6×M4(2)), (D5×C24)⋊16C2, (C4×D5).1C12, (D5×C12).6C4, C4.15(D5×C12), C12.85(C4×D5), C4.Dic5⋊5C6, C60.167(C2×C4), C20.33(C2×C12), (C5×M4(2))⋊3C6, C15⋊27(C2×M4(2)), C22.7(D5×C12), D10.20(C2×C12), (C2×C12).357D10, (C15×M4(2))⋊9C2, C20.37(C22×C6), (C2×Dic5).7C12, (C6×Dic5).16C4, (C22×D5).5C12, C10.28(C22×C12), C30.186(C22×C4), (C2×C60).285C22, Dic5.13(C2×C12), C12.244(C22×D5), (D5×C12).119C22, (C2×C4×D5).3C6, C4.38(D5×C2×C6), (D5×C2×C6).11C4, C2.16(D5×C2×C12), C6.111(C2×C4×D5), C5⋊2C8⋊11(C2×C6), (D5×C2×C12).14C2, (C2×C4).44(C6×D5), (C2×C6).41(C4×D5), (C3×C8⋊D5)⋊13C2, (C2×C20).22(C2×C6), (C6×D5).69(C2×C4), (C4×D5).38(C2×C6), (C2×C30).122(C2×C4), (C2×C10).25(C2×C12), (C3×C5⋊2C8)⋊44C22, (C3×C4.Dic5)⋊17C2, (C3×Dic5).78(C2×C4), SmallGroup(480,699)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D5×M4(2)
G = < a,b,c,d,e | a3=b5=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: 368 in 136 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C6×M4(2), C3×C5⋊2C8, C120, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×M4(2), D5×C24, C3×C8⋊D5, C3×C4.Dic5, C15×M4(2), D5×C2×C12, C3×D5×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, M4(2), C22×C4, D10, C2×C12, C22×C6, C3×D5, C2×M4(2), C4×D5, C22×D5, C3×M4(2), C22×C12, C6×D5, C2×C4×D5, C6×M4(2), D5×C12, D5×C2×C6, D5×M4(2), D5×C2×C12, C3×D5×M4(2)
►On 120 points
Generators in S120
(1 104 64)(2 97 57)(3 98 58)(4 99 59)(5 100 60)(6 101 61)(7 102 62)(8 103 63)(9 86 43)(10 87 44)(11 88 45)(12 81 46)(13 82 47)(14 83 48)(15 84 41)(16 85 42)(17 52 73)(18 53 74)(19 54 75)(20 55 76)(21 56 77)(22 49 78)(23 50 79)(24 51 80)(25 96 113)(26 89 114)(27 90 115)(28 91 116)(29 92 117)(30 93 118)(31 94 119)(32 95 120)(33 111 69)(34 112 70)(35 105 71)(36 106 72)(37 107 65)(38 108 66)(39 109 67)(40 110 68)
(1 108 56 29 44)(2 109 49 30 45)(3 110 50 31 46)(4 111 51 32 47)(5 112 52 25 48)(6 105 53 26 41)(7 106 54 27 42)(8 107 55 28 43)(9 103 65 76 91)(10 104 66 77 92)(11 97 67 78 93)(12 98 68 79 94)(13 99 69 80 95)(14 100 70 73 96)(15 101 71 74 89)(16 102 72 75 90)(17 113 83 60 34)(18 114 84 61 35)(19 115 85 62 36)(20 116 86 63 37)(21 117 87 64 38)(22 118 88 57 39)(23 119 81 58 40)(24 120 82 59 33)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 103)(10 104)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(25 112)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 120)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)
(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)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)(114 118)(116 120)
G:=sub<Sym(120)| (1,104,64)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,101,61)(7,102,62)(8,103,63)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,52,73)(18,53,74)(19,54,75)(20,55,76)(21,56,77)(22,49,78)(23,50,79)(24,51,80)(25,96,113)(26,89,114)(27,90,115)(28,91,116)(29,92,117)(30,93,118)(31,94,119)(32,95,120)(33,111,69)(34,112,70)(35,105,71)(36,106,72)(37,107,65)(38,108,66)(39,109,67)(40,110,68), (1,108,56,29,44)(2,109,49,30,45)(3,110,50,31,46)(4,111,51,32,47)(5,112,52,25,48)(6,105,53,26,41)(7,106,54,27,42)(8,107,55,28,43)(9,103,65,76,91)(10,104,66,77,92)(11,97,67,78,93)(12,98,68,79,94)(13,99,69,80,95)(14,100,70,73,96)(15,101,71,74,89)(16,102,72,75,90)(17,113,83,60,34)(18,114,84,61,35)(19,115,85,62,36)(20,116,86,63,37)(21,117,87,64,38)(22,118,88,57,39)(23,119,81,58,40)(24,120,82,59,33), (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(25,112)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,120)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90), (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)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111)(114,118)(116,120)>;
G:=Group( (1,104,64)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,101,61)(7,102,62)(8,103,63)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,52,73)(18,53,74)(19,54,75)(20,55,76)(21,56,77)(22,49,78)(23,50,79)(24,51,80)(25,96,113)(26,89,114)(27,90,115)(28,91,116)(29,92,117)(30,93,118)(31,94,119)(32,95,120)(33,111,69)(34,112,70)(35,105,71)(36,106,72)(37,107,65)(38,108,66)(39,109,67)(40,110,68), (1,108,56,29,44)(2,109,49,30,45)(3,110,50,31,46)(4,111,51,32,47)(5,112,52,25,48)(6,105,53,26,41)(7,106,54,27,42)(8,107,55,28,43)(9,103,65,76,91)(10,104,66,77,92)(11,97,67,78,93)(12,98,68,79,94)(13,99,69,80,95)(14,100,70,73,96)(15,101,71,74,89)(16,102,72,75,90)(17,113,83,60,34)(18,114,84,61,35)(19,115,85,62,36)(20,116,86,63,37)(21,117,87,64,38)(22,118,88,57,39)(23,119,81,58,40)(24,120,82,59,33), (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(25,112)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,120)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90), (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)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111)(114,118)(116,120) );
G=PermutationGroup([[(1,104,64),(2,97,57),(3,98,58),(4,99,59),(5,100,60),(6,101,61),(7,102,62),(8,103,63),(9,86,43),(10,87,44),(11,88,45),(12,81,46),(13,82,47),(14,83,48),(15,84,41),(16,85,42),(17,52,73),(18,53,74),(19,54,75),(20,55,76),(21,56,77),(22,49,78),(23,50,79),(24,51,80),(25,96,113),(26,89,114),(27,90,115),(28,91,116),(29,92,117),(30,93,118),(31,94,119),(32,95,120),(33,111,69),(34,112,70),(35,105,71),(36,106,72),(37,107,65),(38,108,66),(39,109,67),(40,110,68)], [(1,108,56,29,44),(2,109,49,30,45),(3,110,50,31,46),(4,111,51,32,47),(5,112,52,25,48),(6,105,53,26,41),(7,106,54,27,42),(8,107,55,28,43),(9,103,65,76,91),(10,104,66,77,92),(11,97,67,78,93),(12,98,68,79,94),(13,99,69,80,95),(14,100,70,73,96),(15,101,71,74,89),(16,102,72,75,90),(17,113,83,60,34),(18,114,84,61,35),(19,115,85,62,36),(20,116,86,63,37),(21,117,87,64,38),(22,118,88,57,39),(23,119,81,58,40),(24,120,82,59,33)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,103),(10,104),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(25,112),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,120),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(57,88),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90)], [(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),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(74,78),(76,80),(82,86),(84,88),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111),(114,118),(116,120)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | ··· | 24H | 24I | ··· | 24P | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
120 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 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D5 | M4(2) | D10 | D10 | C3×D5 | C4×D5 | C4×D5 | C3×M4(2) | C6×D5 | C6×D5 | D5×C12 | D5×C12 | D5×M4(2) | C3×D5×M4(2) |
| kernel | C3×D5×M4(2) | D5×C24 | C3×C8⋊D5 | C3×C4.Dic5 | C15×M4(2) | D5×C2×C12 | D5×M4(2) | D5×C12 | C6×Dic5 | D5×C2×C6 | C8×D5 | C8⋊D5 | C4.Dic5 | C5×M4(2) | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C3×M4(2) | C3×D5 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | D5 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C3×D5×M4(2) ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 64 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,225,0,0,0,0,225],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,0,64,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,240] >;
C3×D5×M4(2) in GAP, Magma, Sage, TeX
C_3\times D_5\times M_4(2)
% in TeX
G:=Group("C3xD5xM4(2)"); // GroupNames label
G:=SmallGroup(480,699);
// by ID
G=gap.SmallGroup(480,699);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,555,142,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^5=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