metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D15⋊2C8, D30.4C4, C20.31D6, C12.31D10, C60.31C22, Dic15.4C4, C5⋊3(S3×C8), C3⋊C8⋊6D5, C3⋊1(C8×D5), C15⋊7(C2×C8), C5⋊2C8⋊6S3, C6.1(C4×D5), C10.8(C4×S3), C4.24(S3×D5), C30.24(C2×C4), (C4×D15).4C2, C2.1(D30.C2), (C5×C3⋊C8)⋊5C2, (C3×C5⋊2C8)⋊5C2, SmallGroup(240,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D15⋊2C8 |
Generators and relations for D15⋊2C8
G = < a,b,c | a15=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Smallest permutation representation of D15⋊2C8
►On 120 points
Generators in S120
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 49)(32 48)(33 47)(34 46)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)
(1 118 50 83 16 98 31 75)(2 114 51 79 17 94 32 71)(3 110 52 90 18 105 33 67)(4 106 53 86 19 101 34 63)(5 117 54 82 20 97 35 74)(6 113 55 78 21 93 36 70)(7 109 56 89 22 104 37 66)(8 120 57 85 23 100 38 62)(9 116 58 81 24 96 39 73)(10 112 59 77 25 92 40 69)(11 108 60 88 26 103 41 65)(12 119 46 84 27 99 42 61)(13 115 47 80 28 95 43 72)(14 111 48 76 29 91 44 68)(15 107 49 87 30 102 45 64)
G:=sub<Sym(120)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110), (1,118,50,83,16,98,31,75)(2,114,51,79,17,94,32,71)(3,110,52,90,18,105,33,67)(4,106,53,86,19,101,34,63)(5,117,54,82,20,97,35,74)(6,113,55,78,21,93,36,70)(7,109,56,89,22,104,37,66)(8,120,57,85,23,100,38,62)(9,116,58,81,24,96,39,73)(10,112,59,77,25,92,40,69)(11,108,60,88,26,103,41,65)(12,119,46,84,27,99,42,61)(13,115,47,80,28,95,43,72)(14,111,48,76,29,91,44,68)(15,107,49,87,30,102,45,64)>;
G:=Group( (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110), (1,118,50,83,16,98,31,75)(2,114,51,79,17,94,32,71)(3,110,52,90,18,105,33,67)(4,106,53,86,19,101,34,63)(5,117,54,82,20,97,35,74)(6,113,55,78,21,93,36,70)(7,109,56,89,22,104,37,66)(8,120,57,85,23,100,38,62)(9,116,58,81,24,96,39,73)(10,112,59,77,25,92,40,69)(11,108,60,88,26,103,41,65)(12,119,46,84,27,99,42,61)(13,115,47,80,28,95,43,72)(14,111,48,76,29,91,44,68)(15,107,49,87,30,102,45,64) );
G=PermutationGroup([[(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,49),(32,48),(33,47),(34,46),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(91,109),(92,108),(93,107),(94,106),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110)], [(1,118,50,83,16,98,31,75),(2,114,51,79,17,94,32,71),(3,110,52,90,18,105,33,67),(4,106,53,86,19,101,34,63),(5,117,54,82,20,97,35,74),(6,113,55,78,21,93,36,70),(7,109,56,89,22,104,37,66),(8,120,57,85,23,100,38,62),(9,116,58,81,24,96,39,73),(10,112,59,77,25,92,40,69),(11,108,60,88,26,103,41,65),(12,119,46,84,27,99,42,61),(13,115,47,80,28,95,43,72),(14,111,48,76,29,91,44,68),(15,107,49,87,30,102,45,64)]])
D15⋊2C8 is a maximal subgroup of
D15⋊C16 D30.C8 S3×C8×D5 C40⋊D6 C40.34D6 C40.55D6 D60.5C4 D60.4C4 D15⋊4M4(2) D15⋊D8 Dic10⋊D6 D20.10D6 D30.11D4 D15⋊SD16 D15⋊Q16 D20.16D6 D12.D10
D15⋊2C8 is a maximal quotient of
D15⋊2C16 D30.5C8 Dic15⋊4C8 D30⋊4C8 C60.14Q8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 15 | 15 | 2 | 1 | 1 | 15 | 15 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D5 | D6 | D10 | C4×S3 | C4×D5 | S3×C8 | C8×D5 | S3×D5 | D30.C2 | D15⋊2C8 |
| kernel | D15⋊2C8 | C5×C3⋊C8 | C3×C5⋊2C8 | C4×D15 | Dic15 | D30 | D15 | C5⋊2C8 | C3⋊C8 | C20 | C12 | C10 | C6 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of D15⋊2C8 ►in GL4(𝔽241) generated by
| 190 | 51 | 0 | 0 |
| 190 | 240 | 0 | 0 |
| 0 | 0 | 239 | 49 |
| 0 | 0 | 177 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 192 |
| 0 | 0 | 64 | 239 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 233 | 211 |
G:=sub<GL(4,GF(241))| [190,190,0,0,51,240,0,0,0,0,239,177,0,0,49,1],[0,1,0,0,1,0,0,0,0,0,2,64,0,0,192,239],[1,0,0,0,0,1,0,0,0,0,30,233,0,0,0,211] >;
D15⋊2C8 in GAP, Magma, Sage, TeX
D_{15}\rtimes_2C_8 % in TeX
G:=Group("D15:2C8"); // GroupNames label
G:=SmallGroup(240,9);
// by ID
G=gap.SmallGroup(240,9);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,31,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^15=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations
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