metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D30, C40⋊3D6, C24⋊3D10, D120⋊2C2, C4.14D60, C60.12D4, C120⋊1C22, C12.24D20, C20.24D12, C22.5D60, D60⋊21C22, M4(2)⋊1D15, C60.248C23, Dic30⋊19C22, C5⋊3(C8⋊D6), (C2×C30).5D4, C24⋊D5⋊1C2, (C2×D60)⋊10C2, C3⋊3(C8⋊D10), (C2×C4).14D30, C6.41(C2×D20), C2.15(C2×D60), (C2×C6).10D20, C15⋊25(C8⋊C22), (C2×C10).10D12, C30.269(C2×D4), (C2×C20).142D6, C10.42(C2×D12), (C5×M4(2))⋊1S3, (C3×M4(2))⋊1D5, (C2×C12).141D10, (C15×M4(2))⋊1C2, (C2×C60).68C22, D60⋊11C2⋊12C2, C4.29(C22×D15), C20.219(C22×S3), C12.221(C22×D5), SmallGroup(480,873)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D30
G = < a,b,c | a8=b30=c2=1, bab-1=a5, cac=a3, cbc=b-1 >
Subgroups: 1172 in 136 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C22×S3, D15, C30, C30, C8⋊C22, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C22×D5, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, Dic15, C60, D30, C2×C30, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, C8⋊D6, C120, Dic30, C4×D15, D60, D60, D60, C15⋊7D4, C2×C60, C22×D15, C8⋊D10, C24⋊D5, D120, C15×M4(2), C2×D60, D60⋊11C2, C8⋊D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, D15, C8⋊C22, D20, C22×D5, C2×D12, D30, C2×D20, C8⋊D6, D60, C22×D15, C8⋊D10, C2×D60, C8⋊D30
►On 120 points
(1 91 41 86 16 106 46 71)(2 107 42 72 17 92 47 87)(3 93 43 88 18 108 48 73)(4 109 44 74 19 94 49 89)(5 95 45 90 20 110 50 75)(6 111 31 76 21 96 51 61)(7 97 32 62 22 112 52 77)(8 113 33 78 23 98 53 63)(9 99 34 64 24 114 54 79)(10 115 35 80 25 100 55 65)(11 101 36 66 26 116 56 81)(12 117 37 82 27 102 57 67)(13 103 38 68 28 118 58 83)(14 119 39 84 29 104 59 69)(15 105 40 70 30 120 60 85)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)
G:=sub<Sym(120)| (1,91,41,86,16,106,46,71)(2,107,42,72,17,92,47,87)(3,93,43,88,18,108,48,73)(4,109,44,74,19,94,49,89)(5,95,45,90,20,110,50,75)(6,111,31,76,21,96,51,61)(7,97,32,62,22,112,52,77)(8,113,33,78,23,98,53,63)(9,99,34,64,24,114,54,79)(10,115,35,80,25,100,55,65)(11,101,36,66,26,116,56,81)(12,117,37,82,27,102,57,67)(13,103,38,68,28,118,58,83)(14,119,39,84,29,104,59,69)(15,105,40,70,30,120,60,85), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)>;
G:=Group( (1,91,41,86,16,106,46,71)(2,107,42,72,17,92,47,87)(3,93,43,88,18,108,48,73)(4,109,44,74,19,94,49,89)(5,95,45,90,20,110,50,75)(6,111,31,76,21,96,51,61)(7,97,32,62,22,112,52,77)(8,113,33,78,23,98,53,63)(9,99,34,64,24,114,54,79)(10,115,35,80,25,100,55,65)(11,101,36,66,26,116,56,81)(12,117,37,82,27,102,57,67)(13,103,38,68,28,118,58,83)(14,119,39,84,29,104,59,69)(15,105,40,70,30,120,60,85), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101) );
G=PermutationGroup([[(1,91,41,86,16,106,46,71),(2,107,42,72,17,92,47,87),(3,93,43,88,18,108,48,73),(4,109,44,74,19,94,49,89),(5,95,45,90,20,110,50,75),(6,111,31,76,21,96,51,61),(7,97,32,62,22,112,52,77),(8,113,33,78,23,98,53,63),(9,99,34,64,24,114,54,79),(10,115,35,80,25,100,55,65),(11,101,36,66,26,116,56,81),(12,117,37,82,27,102,57,67),(13,103,38,68,28,118,58,83),(14,119,39,84,29,104,59,69),(15,105,40,70,30,120,60,85)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,60),(42,59),(43,58),(44,57),(45,56),(61,100),(62,99),(63,98),(64,97),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 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 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 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 | 60 | 60 | 60 | 2 | 2 | 2 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 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 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D12 | D15 | D20 | D20 | D30 | D30 | D60 | D60 | C8⋊C22 | C8⋊D6 | C8⋊D10 | C8⋊D30 |
| kernel | C8⋊D30 | C24⋊D5 | D120 | C15×M4(2) | C2×D60 | D60⋊11C2 | C5×M4(2) | C60 | C2×C30 | C3×M4(2) | C40 | C2×C20 | 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 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C8⋊D30 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 92 | 157 |
| 0 | 0 | 0 | 1 | 84 | 146 |
| 0 | 0 | 161 | 91 | 240 | 0 |
| 0 | 0 | 150 | 223 | 0 | 240 |
| 208 | 187 | 0 | 0 | 0 | 0 |
| 177 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 41 | 17 | 190 | 1 |
| 0 | 0 | 224 | 185 | 240 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 30 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 131 | 147 | 0 | 0 |
| 0 | 0 | 80 | 110 | 0 | 0 |
| 0 | 0 | 108 | 80 | 78 | 197 |
| 0 | 0 | 45 | 133 | 78 | 163 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,161,150,0,0,0,1,91,223,0,0,92,84,240,0,0,0,157,146,0,240],[208,177,0,0,0,0,187,34,0,0,0,0,0,0,51,1,41,224,0,0,240,0,17,185,0,0,0,0,190,240,0,0,0,0,1,0],[1,30,0,0,0,0,0,240,0,0,0,0,0,0,131,80,108,45,0,0,147,110,80,133,0,0,0,0,78,78,0,0,0,0,197,163] >;
C8⋊D30 in GAP, Magma, Sage, TeX
C_8\rtimes D_{30} % in TeX
G:=Group("C8:D30"); // GroupNames label
G:=SmallGroup(480,873);
// by ID
G=gap.SmallGroup(480,873);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^30=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations