direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.46D4, C60.224D4, C20.63D12, C4.11(C5×D12), C12.46(C5×D4), (C22×S3).C20, (C2×D12).6C10, (C2×C20).213D6, C15⋊9(C4.D4), M4(2)⋊3(C5×S3), (C5×M4(2))⋊7S3, C4.Dic3⋊2C10, C22.4(S3×C20), C10.55(D6⋊C4), (C10×D12).16C2, (C3×M4(2))⋊7C10, C20.114(C3⋊D4), C30.97(C22⋊C4), (C15×M4(2))⋊17C2, (C2×C60).342C22, (S3×C2×C10).7C4, C2.9(C5×D6⋊C4), C3⋊1(C5×C4.D4), (C2×C6).2(C2×C20), (C2×C4).1(S3×C10), C4.21(C5×C3⋊D4), C6.8(C5×C22⋊C4), (C2×C10).63(C4×S3), (C2×C30).126(C2×C4), (C2×C12).12(C2×C10), (C5×C4.Dic3)⋊14C2, SmallGroup(480,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.46D4
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >
Subgroups: 292 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, D6, C2×C6, C15, M4(2), M4(2), C2×D4, C20, C2×C10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, C5×S3, C30, C30, C4.D4, C40, C2×C20, C5×D4, C22×C10, C4.Dic3, C3×M4(2), C2×D12, C60, S3×C10, C2×C30, C5×M4(2), C5×M4(2), D4×C10, C12.46D4, C5×C3⋊C8, C120, C5×D12, C2×C60, S3×C2×C10, C5×C4.D4, C5×C4.Dic3, C15×M4(2), C10×D12, C5×C12.46D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4.D4, C2×C20, C5×D4, D6⋊C4, S3×C10, C5×C22⋊C4, C12.46D4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4.D4, C5×D6⋊C4, C5×C12.46D4
►On 120 points
Generators in S120
(1 15 35 108 29)(2 16 36 109 30)(3 9 37 110 31)(4 10 38 111 32)(5 11 39 112 25)(6 12 40 105 26)(7 13 33 106 27)(8 14 34 107 28)(17 43 58 82 51)(18 44 59 83 52)(19 45 60 84 53)(20 46 61 85 54)(21 47 62 86 55)(22 48 63 87 56)(23 41 64 88 49)(24 42 57 81 50)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(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)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)
(1 99 19)(2 100 20)(3 101 21)(4 102 22)(5 103 23)(6 104 24)(7 97 17)(8 98 18)(9 72 47)(10 65 48)(11 66 41)(12 67 42)(13 68 43)(14 69 44)(15 70 45)(16 71 46)(25 78 49)(26 79 50)(27 80 51)(28 73 52)(29 74 53)(30 75 54)(31 76 55)(32 77 56)(33 96 58)(34 89 59)(35 90 60)(36 91 61)(37 92 62)(38 93 63)(39 94 64)(40 95 57)(81 105 120)(82 106 113)(83 107 114)(84 108 115)(85 109 116)(86 110 117)(87 111 118)(88 112 119)
(2 6)(3 7)(9 13)(12 16)(17 101)(18 98)(19 99)(20 104)(21 97)(22 102)(23 103)(24 100)(26 30)(27 31)(33 37)(36 40)(41 66)(42 71)(43 72)(44 69)(45 70)(46 67)(47 68)(48 65)(49 78)(50 75)(51 76)(52 73)(53 74)(54 79)(55 80)(56 77)(57 91)(58 92)(59 89)(60 90)(61 95)(62 96)(63 93)(64 94)(81 116)(82 117)(83 114)(84 115)(85 120)(86 113)(87 118)(88 119)(105 109)(106 110)
G:=sub<Sym(120)| (1,15,35,108,29)(2,16,36,109,30)(3,9,37,110,31)(4,10,38,111,32)(5,11,39,112,25)(6,12,40,105,26)(7,13,33,106,27)(8,14,34,107,28)(17,43,58,82,51)(18,44,59,83,52)(19,45,60,84,53)(20,46,61,85,54)(21,47,62,86,55)(22,48,63,87,56)(23,41,64,88,49)(24,42,57,81,50)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120), (1,99,19)(2,100,20)(3,101,21)(4,102,22)(5,103,23)(6,104,24)(7,97,17)(8,98,18)(9,72,47)(10,65,48)(11,66,41)(12,67,42)(13,68,43)(14,69,44)(15,70,45)(16,71,46)(25,78,49)(26,79,50)(27,80,51)(28,73,52)(29,74,53)(30,75,54)(31,76,55)(32,77,56)(33,96,58)(34,89,59)(35,90,60)(36,91,61)(37,92,62)(38,93,63)(39,94,64)(40,95,57)(81,105,120)(82,106,113)(83,107,114)(84,108,115)(85,109,116)(86,110,117)(87,111,118)(88,112,119), (2,6)(3,7)(9,13)(12,16)(17,101)(18,98)(19,99)(20,104)(21,97)(22,102)(23,103)(24,100)(26,30)(27,31)(33,37)(36,40)(41,66)(42,71)(43,72)(44,69)(45,70)(46,67)(47,68)(48,65)(49,78)(50,75)(51,76)(52,73)(53,74)(54,79)(55,80)(56,77)(57,91)(58,92)(59,89)(60,90)(61,95)(62,96)(63,93)(64,94)(81,116)(82,117)(83,114)(84,115)(85,120)(86,113)(87,118)(88,119)(105,109)(106,110)>;
G:=Group( (1,15,35,108,29)(2,16,36,109,30)(3,9,37,110,31)(4,10,38,111,32)(5,11,39,112,25)(6,12,40,105,26)(7,13,33,106,27)(8,14,34,107,28)(17,43,58,82,51)(18,44,59,83,52)(19,45,60,84,53)(20,46,61,85,54)(21,47,62,86,55)(22,48,63,87,56)(23,41,64,88,49)(24,42,57,81,50)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(105,109)(107,111)(114,118)(116,120), (1,99,19)(2,100,20)(3,101,21)(4,102,22)(5,103,23)(6,104,24)(7,97,17)(8,98,18)(9,72,47)(10,65,48)(11,66,41)(12,67,42)(13,68,43)(14,69,44)(15,70,45)(16,71,46)(25,78,49)(26,79,50)(27,80,51)(28,73,52)(29,74,53)(30,75,54)(31,76,55)(32,77,56)(33,96,58)(34,89,59)(35,90,60)(36,91,61)(37,92,62)(38,93,63)(39,94,64)(40,95,57)(81,105,120)(82,106,113)(83,107,114)(84,108,115)(85,109,116)(86,110,117)(87,111,118)(88,112,119), (2,6)(3,7)(9,13)(12,16)(17,101)(18,98)(19,99)(20,104)(21,97)(22,102)(23,103)(24,100)(26,30)(27,31)(33,37)(36,40)(41,66)(42,71)(43,72)(44,69)(45,70)(46,67)(47,68)(48,65)(49,78)(50,75)(51,76)(52,73)(53,74)(54,79)(55,80)(56,77)(57,91)(58,92)(59,89)(60,90)(61,95)(62,96)(63,93)(64,94)(81,116)(82,117)(83,114)(84,115)(85,120)(86,113)(87,118)(88,119)(105,109)(106,110) );
G=PermutationGroup([[(1,15,35,108,29),(2,16,36,109,30),(3,9,37,110,31),(4,10,38,111,32),(5,11,39,112,25),(6,12,40,105,26),(7,13,33,106,27),(8,14,34,107,28),(17,43,58,82,51),(18,44,59,83,52),(19,45,60,84,53),(20,46,61,85,54),(21,47,62,86,55),(22,48,63,87,56),(23,41,64,88,49),(24,42,57,81,50),(65,93,118,77,102),(66,94,119,78,103),(67,95,120,79,104),(68,96,113,80,97),(69,89,114,73,98),(70,90,115,74,99),(71,91,116,75,100),(72,92,117,76,101)], [(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),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79),(81,85),(83,87),(89,93),(91,95),(98,102),(100,104),(105,109),(107,111),(114,118),(116,120)], [(1,99,19),(2,100,20),(3,101,21),(4,102,22),(5,103,23),(6,104,24),(7,97,17),(8,98,18),(9,72,47),(10,65,48),(11,66,41),(12,67,42),(13,68,43),(14,69,44),(15,70,45),(16,71,46),(25,78,49),(26,79,50),(27,80,51),(28,73,52),(29,74,53),(30,75,54),(31,76,55),(32,77,56),(33,96,58),(34,89,59),(35,90,60),(36,91,61),(37,92,62),(38,93,63),(39,94,64),(40,95,57),(81,105,120),(82,106,113),(83,107,114),(84,108,115),(85,109,116),(86,110,117),(87,111,118),(88,112,119)], [(2,6),(3,7),(9,13),(12,16),(17,101),(18,98),(19,99),(20,104),(21,97),(22,102),(23,103),(24,100),(26,30),(27,31),(33,37),(36,40),(41,66),(42,71),(43,72),(44,69),(45,70),(46,67),(47,68),(48,65),(49,78),(50,75),(51,76),(52,73),(53,74),(54,79),(55,80),(56,77),(57,91),(58,92),(59,89),(60,90),(61,95),(62,96),(63,93),(64,94),(81,116),(82,117),(83,114),(84,115),(85,120),(86,113),(87,118),(88,119),(105,109),(106,110)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 40I | ··· | 40P | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 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 | 12 | 12 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | C5×S3 | C5×D4 | S3×C10 | C5×D12 | C5×C3⋊D4 | S3×C20 | C4.D4 | C12.46D4 | C5×C4.D4 | C5×C12.46D4 |
| kernel | C5×C12.46D4 | C5×C4.Dic3 | C15×M4(2) | C10×D12 | S3×C2×C10 | C12.46D4 | C4.Dic3 | C3×M4(2) | C2×D12 | C22×S3 | C5×M4(2) | C60 | C2×C20 | C20 | C20 | C2×C10 | M4(2) | C12 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C12.46D4 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 90 | 81 | 226 | 153 |
| 205 | 106 | 120 | 34 |
| 198 | 86 | 47 | 16 |
| 155 | 43 | 29 | 239 |
| 1 | 0 | 125 | 189 |
| 0 | 1 | 72 | 8 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 240 | 1 | 10 | 88 |
| 240 | 0 | 98 | 231 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 1 | 240 | 205 | 127 |
| 0 | 240 | 169 | 36 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[90,205,198,155,81,106,86,43,226,120,47,29,153,34,16,239],[1,0,0,0,0,1,0,0,125,72,240,0,189,8,0,240],[240,240,0,0,1,0,0,0,10,98,240,240,88,231,1,0],[1,0,0,0,240,240,0,0,205,169,1,0,127,36,240,240] >;
C5×C12.46D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}._{46}D_4 % in TeX
G:=Group("C5xC12.46D4"); // GroupNames label
G:=SmallGroup(480,142);
// by ID
G=gap.SmallGroup(480,142);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,136,2111,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations