metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D15⋊1M4(2), C5⋊C8⋊2D6, C4.F5⋊5S3, D15⋊C8⋊2C2, D6.F5⋊3C2, (C4×S3).2F5, D6.7(C2×F5), C5⋊2(S3×M4(2)), C4.21(S3×F5), (S3×C20).2C4, C20.21(C4×S3), C60.21(C2×C4), (C4×D15).2C4, D30.9(C2×C4), (C4×D5).66D6, C15⋊2(C2×M4(2)), C12.17(C2×F5), C12.F5⋊6C2, C15⋊C8⋊2C22, D10.22(C4×S3), C3⋊2(D5⋊M4(2)), (D5×Dic3).6C4, C6.10(C22×F5), C30.10(C22×C4), Dic3.13(C2×F5), Dic15.11(C2×C4), (D5×C12).52C22, D30.C2.14C22, (S3×Dic5).14C22, Dic5.30(C22×S3), (C3×Dic5).28C23, (C4×S3×D5).4C2, (C2×S3×D5).6C4, C2.14(C2×S3×F5), (C3×C5⋊C8)⋊2C22, C10.10(S3×C2×C4), (C3×C4.F5)⋊6C2, (S3×C10).9(C2×C4), (C6×D5).19(C2×C4), (C5×Dic3).11(C2×C4), SmallGroup(480,991)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D15⋊M4(2)
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a7, dad=a4, cbc-1=a6b, dbd=a3b, dcd=c5 >
Subgroups: 676 in 136 conjugacy classes, 48 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, C4.F5, C4.F5, C22.F5, C2×C4×D5, S3×M4(2), C3×C5⋊C8, C15⋊C8, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5⋊M4(2), D15⋊C8, D6.F5, C3×C4.F5, C12.F5, C4×S3×D5, D15⋊M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, S3×C2×C4, C22×F5, S3×M4(2), S3×F5, D5⋊M4(2), C2×S3×F5, D15⋊M4(2)
►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 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 120)(105 119)
(1 104 31 81 16 106 53 61)(2 102 35 88 17 119 57 68)(3 100 39 80 18 117 46 75)(4 98 43 87 19 115 50 67)(5 96 32 79 20 113 54 74)(6 94 36 86 21 111 58 66)(7 92 40 78 22 109 47 73)(8 105 44 85 23 107 51 65)(9 103 33 77 24 120 55 72)(10 101 37 84 25 118 59 64)(11 99 41 76 26 116 48 71)(12 97 45 83 27 114 52 63)(13 95 34 90 28 112 56 70)(14 93 38 82 29 110 60 62)(15 91 42 89 30 108 49 69)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(17 20)(18 24)(19 28)(22 25)(23 29)(27 30)(32 35)(33 39)(34 43)(37 40)(38 44)(42 45)(46 55)(47 59)(49 52)(50 56)(51 60)(54 57)(61 81)(62 85)(63 89)(64 78)(65 82)(66 86)(67 90)(68 79)(69 83)(70 87)(71 76)(72 80)(73 84)(74 88)(75 77)(91 114)(92 118)(93 107)(94 111)(95 115)(96 119)(97 108)(98 112)(99 116)(100 120)(101 109)(102 113)(103 117)(104 106)(105 110)
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)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,120)(105,119), (1,104,31,81,16,106,53,61)(2,102,35,88,17,119,57,68)(3,100,39,80,18,117,46,75)(4,98,43,87,19,115,50,67)(5,96,32,79,20,113,54,74)(6,94,36,86,21,111,58,66)(7,92,40,78,22,109,47,73)(8,105,44,85,23,107,51,65)(9,103,33,77,24,120,55,72)(10,101,37,84,25,118,59,64)(11,99,41,76,26,116,48,71)(12,97,45,83,27,114,52,63)(13,95,34,90,28,112,56,70)(14,93,38,82,29,110,60,62)(15,91,42,89,30,108,49,69), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(17,20)(18,24)(19,28)(22,25)(23,29)(27,30)(32,35)(33,39)(34,43)(37,40)(38,44)(42,45)(46,55)(47,59)(49,52)(50,56)(51,60)(54,57)(61,81)(62,85)(63,89)(64,78)(65,82)(66,86)(67,90)(68,79)(69,83)(70,87)(71,76)(72,80)(73,84)(74,88)(75,77)(91,114)(92,118)(93,107)(94,111)(95,115)(96,119)(97,108)(98,112)(99,116)(100,120)(101,109)(102,113)(103,117)(104,106)(105,110)>;
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)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,120)(105,119), (1,104,31,81,16,106,53,61)(2,102,35,88,17,119,57,68)(3,100,39,80,18,117,46,75)(4,98,43,87,19,115,50,67)(5,96,32,79,20,113,54,74)(6,94,36,86,21,111,58,66)(7,92,40,78,22,109,47,73)(8,105,44,85,23,107,51,65)(9,103,33,77,24,120,55,72)(10,101,37,84,25,118,59,64)(11,99,41,76,26,116,48,71)(12,97,45,83,27,114,52,63)(13,95,34,90,28,112,56,70)(14,93,38,82,29,110,60,62)(15,91,42,89,30,108,49,69), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(17,20)(18,24)(19,28)(22,25)(23,29)(27,30)(32,35)(33,39)(34,43)(37,40)(38,44)(42,45)(46,55)(47,59)(49,52)(50,56)(51,60)(54,57)(61,81)(62,85)(63,89)(64,78)(65,82)(66,86)(67,90)(68,79)(69,83)(70,87)(71,76)(72,80)(73,84)(74,88)(75,77)(91,114)(92,118)(93,107)(94,111)(95,115)(96,119)(97,108)(98,112)(99,116)(100,120)(101,109)(102,113)(103,117)(104,106)(105,110) );
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),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53),(61,80),(62,79),(63,78),(64,77),(65,76),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,120),(105,119)], [(1,104,31,81,16,106,53,61),(2,102,35,88,17,119,57,68),(3,100,39,80,18,117,46,75),(4,98,43,87,19,115,50,67),(5,96,32,79,20,113,54,74),(6,94,36,86,21,111,58,66),(7,92,40,78,22,109,47,73),(8,105,44,85,23,107,51,65),(9,103,33,77,24,120,55,72),(10,101,37,84,25,118,59,64),(11,99,41,76,26,116,48,71),(12,97,45,83,27,114,52,63),(13,95,34,90,28,112,56,70),(14,93,38,82,29,110,60,62),(15,91,42,89,30,108,49,69)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15),(17,20),(18,24),(19,28),(22,25),(23,29),(27,30),(32,35),(33,39),(34,43),(37,40),(38,44),(42,45),(46,55),(47,59),(49,52),(50,56),(51,60),(54,57),(61,81),(62,85),(63,89),(64,78),(65,82),(66,86),(67,90),(68,79),(69,83),(70,87),(71,76),(72,80),(73,84),(74,88),(75,77),(91,114),(92,118),(93,107),(94,111),(95,115),(96,119),(97,108),(98,112),(99,116),(100,120),(101,109),(102,113),(103,117),(104,106),(105,110)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 6 | 10 | 15 | 15 | 2 | 2 | 3 | 3 | 5 | 5 | 30 | 4 | 2 | 20 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 12 | 12 | 4 | 10 | 10 | 8 | 4 | 4 | 12 | 12 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | F5 | C2×F5 | C2×F5 | C2×F5 | S3×M4(2) | D5⋊M4(2) | S3×F5 | C2×S3×F5 | D15⋊M4(2) |
| kernel | D15⋊M4(2) | D15⋊C8 | D6.F5 | C3×C4.F5 | C12.F5 | C4×S3×D5 | D5×Dic3 | S3×C20 | C4×D15 | C2×S3×D5 | C4.F5 | C5⋊C8 | C4×D5 | D15 | C20 | D10 | C4×S3 | Dic3 | C12 | D6 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of D15⋊M4(2) ►in GL6(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 189 | 0 | 0 |
| 0 | 0 | 52 | 240 | 0 | 0 |
| 0 | 0 | 184 | 15 | 0 | 240 |
| 0 | 0 | 226 | 15 | 1 | 189 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 1 |
| 0 | 0 | 0 | 0 | 189 | 52 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 136 | 104 | 137 |
| 0 | 0 | 53 | 188 | 2 | 137 |
| 0 | 0 | 47 | 72 | 211 | 219 |
| 0 | 0 | 0 | 119 | 158 | 83 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 52 | 0 | 0 |
| 0 | 0 | 240 | 52 | 0 | 0 |
| 0 | 0 | 15 | 79 | 52 | 189 |
| 0 | 0 | 68 | 173 | 1 | 189 |
G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,52,52,184,226,0,0,189,240,15,15,0,0,0,0,0,1,0,0,0,0,240,189],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,189,189,0,0,0,0,1,52],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,53,47,0,0,0,136,188,72,119,0,0,104,2,211,158,0,0,137,137,219,83],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,189,240,15,68,0,0,52,52,79,173,0,0,0,0,52,1,0,0,0,0,189,189] >;
D15⋊M4(2) in GAP, Magma, Sage, TeX
D_{15}\rtimes M_4(2) % in TeX
G:=Group("D15:M4(2)"); // GroupNames label
G:=SmallGroup(480,991);
// by ID
G=gap.SmallGroup(480,991);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,100,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^4,c*b*c^-1=a^6*b,d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations