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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 79)(62 78)(63 77)(64 76)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(91 107)(92 106)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)
(1 115 58 79 20 99 35 62)(2 111 59 90 21 95 36 73)(3 107 60 86 22 91 37 69)(4 118 46 82 23 102 38 65)(5 114 47 78 24 98 39 61)(6 110 48 89 25 94 40 72)(7 106 49 85 26 105 41 68)(8 117 50 81 27 101 42 64)(9 113 51 77 28 97 43 75)(10 109 52 88 29 93 44 71)(11 120 53 84 30 104 45 67)(12 116 54 80 16 100 31 63)(13 112 55 76 17 96 32 74)(14 108 56 87 18 92 33 70)(15 119 57 83 19 103 34 66)
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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,79)(62,78)(63,77)(64,76)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108), (1,115,58,79,20,99,35,62)(2,111,59,90,21,95,36,73)(3,107,60,86,22,91,37,69)(4,118,46,82,23,102,38,65)(5,114,47,78,24,98,39,61)(6,110,48,89,25,94,40,72)(7,106,49,85,26,105,41,68)(8,117,50,81,27,101,42,64)(9,113,51,77,28,97,43,75)(10,109,52,88,29,93,44,71)(11,120,53,84,30,104,45,67)(12,116,54,80,16,100,31,63)(13,112,55,76,17,96,32,74)(14,108,56,87,18,92,33,70)(15,119,57,83,19,103,34,66)>;
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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,79)(62,78)(63,77)(64,76)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108), (1,115,58,79,20,99,35,62)(2,111,59,90,21,95,36,73)(3,107,60,86,22,91,37,69)(4,118,46,82,23,102,38,65)(5,114,47,78,24,98,39,61)(6,110,48,89,25,94,40,72)(7,106,49,85,26,105,41,68)(8,117,50,81,27,101,42,64)(9,113,51,77,28,97,43,75)(10,109,52,88,29,93,44,71)(11,120,53,84,30,104,45,67)(12,116,54,80,16,100,31,63)(13,112,55,76,17,96,32,74)(14,108,56,87,18,92,33,70)(15,119,57,83,19,103,34,66) );
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)], [(1,19),(2,18),(3,17),(4,16),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,79),(62,78),(63,77),(64,76),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(91,107),(92,106),(93,120),(94,119),(95,118),(96,117),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108)], [(1,115,58,79,20,99,35,62),(2,111,59,90,21,95,36,73),(3,107,60,86,22,91,37,69),(4,118,46,82,23,102,38,65),(5,114,47,78,24,98,39,61),(6,110,48,89,25,94,40,72),(7,106,49,85,26,105,41,68),(8,117,50,81,27,101,42,64),(9,113,51,77,28,97,43,75),(10,109,52,88,29,93,44,71),(11,120,53,84,30,104,45,67),(12,116,54,80,16,100,31,63),(13,112,55,76,17,96,32,74),(14,108,56,87,18,92,33,70),(15,119,57,83,19,103,34,66)])
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
Export