direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C8⋊D10, D40⋊2C6, C24⋊16D10, C12.66D20, C60.122D4, C120⋊16C22, C60.264C23, C8⋊1(C6×D5), C40⋊1(C2×C6), C4○D20⋊2C6, (C2×D20)⋊7C6, D20⋊4(C2×C6), C40⋊C2⋊1C6, (C3×D40)⋊10C2, (C6×D20)⋊23C2, C10.11(C6×D4), C20.12(C3×D4), C6.84(C2×D20), C2.15(C6×D20), (C2×C30).82D4, C4.14(C3×D20), (C2×C6).27D20, C15⋊26(C8⋊C22), Dic10⋊4(C2×C6), C30.285(C2×D4), (C5×M4(2))⋊1C6, (C3×M4(2))⋊3D5, M4(2)⋊1(C3×D5), C22.5(C3×D20), (C2×C12).237D10, (C3×D20)⋊34C22, (C15×M4(2))⋊3C2, C20.31(C22×C6), (C2×C60).287C22, C12.237(C22×D5), (C3×Dic10)⋊31C22, C5⋊1(C3×C8⋊C22), C4.30(D5×C2×C6), (C3×C40⋊C2)⋊5C2, (C2×C4).11(C6×D5), (C2×C10).5(C3×D4), (C3×C4○D20)⋊12C2, (C2×C20).24(C2×C6), SmallGroup(480,701)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊D10
G = < a,b,c,d | a3=b8=c10=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 608 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C40, Dic10, C4×D5, D20, D20, D20, C5⋊D4, C2×C20, C22×D5, C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, C3×C8⋊C22, C120, C3×Dic10, D5×C12, C3×D20, C3×D20, C3×D20, C3×C5⋊D4, C2×C60, D5×C2×C6, C8⋊D10, C3×C40⋊C2, C3×D40, C15×M4(2), C6×D20, C3×C4○D20, C3×C8⋊D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×C8⋊C22, C3×D20, D5×C2×C6, C8⋊D10, C6×D20, C3×C8⋊D10
►On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 49 29)(7 50 30)(8 46 26)(9 47 27)(10 48 28)(11 55 35)(12 51 31)(13 52 32)(14 53 33)(15 54 34)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 107 87)(62 108 88)(63 109 89)(64 110 90)(65 101 81)(66 102 82)(67 103 83)(68 104 84)(69 105 85)(70 106 86)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 72 16 66 8 77 12 61)(2 78 17 62 9 73 13 67)(3 74 18 68 10 79 14 63)(4 80 19 64 6 75 15 69)(5 76 20 70 7 71 11 65)(21 92 36 82 26 97 31 87)(22 98 37 88 27 93 32 83)(23 94 38 84 28 99 33 89)(24 100 39 90 29 95 34 85)(25 96 40 86 30 91 35 81)(41 112 56 102 46 117 51 107)(42 118 57 108 47 113 52 103)(43 114 58 104 48 119 53 109)(44 120 59 110 49 115 54 105)(45 116 60 106 50 111 55 101)
(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)
(1 5)(2 4)(6 9)(7 8)(11 16)(12 20)(13 19)(14 18)(15 17)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(81 92)(82 91)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(101 112)(102 111)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)
G:=sub<Sym(120)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,49,29)(7,50,30)(8,46,26)(9,47,27)(10,48,28)(11,55,35)(12,51,31)(13,52,32)(14,53,33)(15,54,34)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,107,87)(62,108,88)(63,109,89)(64,110,90)(65,101,81)(66,102,82)(67,103,83)(68,104,84)(69,105,85)(70,106,86)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,72,16,66,8,77,12,61)(2,78,17,62,9,73,13,67)(3,74,18,68,10,79,14,63)(4,80,19,64,6,75,15,69)(5,76,20,70,7,71,11,65)(21,92,36,82,26,97,31,87)(22,98,37,88,27,93,32,83)(23,94,38,84,28,99,33,89)(24,100,39,90,29,95,34,85)(25,96,40,86,30,91,35,81)(41,112,56,102,46,117,51,107)(42,118,57,108,47,113,52,103)(43,114,58,104,48,119,53,109)(44,120,59,110,49,115,54,105)(45,116,60,106,50,111,55,101), (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), (1,5)(2,4)(6,9)(7,8)(11,16)(12,20)(13,19)(14,18)(15,17)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(81,92)(82,91)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,49,29)(7,50,30)(8,46,26)(9,47,27)(10,48,28)(11,55,35)(12,51,31)(13,52,32)(14,53,33)(15,54,34)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,107,87)(62,108,88)(63,109,89)(64,110,90)(65,101,81)(66,102,82)(67,103,83)(68,104,84)(69,105,85)(70,106,86)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,72,16,66,8,77,12,61)(2,78,17,62,9,73,13,67)(3,74,18,68,10,79,14,63)(4,80,19,64,6,75,15,69)(5,76,20,70,7,71,11,65)(21,92,36,82,26,97,31,87)(22,98,37,88,27,93,32,83)(23,94,38,84,28,99,33,89)(24,100,39,90,29,95,34,85)(25,96,40,86,30,91,35,81)(41,112,56,102,46,117,51,107)(42,118,57,108,47,113,52,103)(43,114,58,104,48,119,53,109)(44,120,59,110,49,115,54,105)(45,116,60,106,50,111,55,101), (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), (1,5)(2,4)(6,9)(7,8)(11,16)(12,20)(13,19)(14,18)(15,17)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(81,92)(82,91)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113) );
G=PermutationGroup([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,49,29),(7,50,30),(8,46,26),(9,47,27),(10,48,28),(11,55,35),(12,51,31),(13,52,32),(14,53,33),(15,54,34),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,107,87),(62,108,88),(63,109,89),(64,110,90),(65,101,81),(66,102,82),(67,103,83),(68,104,84),(69,105,85),(70,106,86),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(1,72,16,66,8,77,12,61),(2,78,17,62,9,73,13,67),(3,74,18,68,10,79,14,63),(4,80,19,64,6,75,15,69),(5,76,20,70,7,71,11,65),(21,92,36,82,26,97,31,87),(22,98,37,88,27,93,32,83),(23,94,38,84,28,99,33,89),(24,100,39,90,29,95,34,85),(25,96,40,86,30,91,35,81),(41,112,56,102,46,117,51,107),(42,118,57,108,47,113,52,103),(43,114,58,104,48,119,53,109),(44,120,59,110,49,115,54,105),(45,116,60,106,50,111,55,101)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,16),(12,20),(13,19),(14,18),(15,17),(21,25),(22,24),(26,30),(27,29),(31,40),(32,39),(33,38),(34,37),(35,36),(41,45),(42,44),(46,50),(47,49),(51,60),(52,59),(53,58),(54,57),(55,56),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77),(81,92),(82,91),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(101,112),(102,111),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 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 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 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 | 20 | 20 | 20 | 1 | 1 | 2 | 2 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 20 | ··· | 20 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | D20 | D20 | C6×D5 | C6×D5 | C3×D20 | C3×D20 | C8⋊C22 | C3×C8⋊C22 | C8⋊D10 | C3×C8⋊D10 |
| kernel | C3×C8⋊D10 | C3×C40⋊C2 | C3×D40 | C15×M4(2) | C6×D20 | C3×C4○D20 | C8⋊D10 | C40⋊C2 | D40 | C5×M4(2) | C2×D20 | C4○D20 | C60 | C2×C30 | C3×M4(2) | C24 | C2×C12 | C20 | C2×C10 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C8⋊D10 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 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 |
| 44 | 156 | 0 | 0 | 0 | 0 |
| 88 | 197 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 197 | 85 | 0 | 0 |
| 0 | 0 | 153 | 44 | 0 | 0 |
| 190 | 52 | 0 | 0 | 0 | 0 |
| 190 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 189 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 52 |
| 0 | 0 | 0 | 0 | 190 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 197 | 200 |
| 0 | 0 | 0 | 0 | 153 | 44 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,88,0,0,0,0,156,197,0,0,0,0,0,0,0,0,197,153,0,0,0,0,85,44,0,0,1,0,0,0,0,0,0,1,0,0],[190,190,0,0,0,0,52,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,197,153,0,0,0,0,200,44] >;
C3×C8⋊D10 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes D_{10} % in TeX
G:=Group("C3xC8:D10"); // GroupNames label
G:=SmallGroup(480,701);
// by ID
G=gap.SmallGroup(480,701);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,142,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^10=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations