Copied to
clipboard

## G = M4(2)⋊D15order 480 = 25·3·5

### 3rd semidirect product of M4(2) and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — M4(2)⋊D15
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C2×D60 — M4(2)⋊D15
 Lower central C15 — C30 — C2×C30 — M4(2)⋊D15
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for M4(2)⋊D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 788 in 92 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, M4(2), M4(2), C2×D4, C20, D10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, D15, C30, C30, C4.D4, C52C8, C40, D20, C2×C20, C22×D5, C4.Dic3, C3×M4(2), C2×D12, C60, D30, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C153C8, C120, D60, C2×C60, C22×D15, C20.46D4, C60.7C4, C15×M4(2), C2×D60, M4(2)⋊D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4.D4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C12.46D4, C4×D15, D60, C157D4, C20.46D4, D303C4, M4(2)⋊D15

Smallest permutation representation of M4(2)⋊D15
On 120 points
Generators in S120
```(1 91 39 80 16 107 47 73)(2 92 40 81 17 108 48 74)(3 93 41 82 18 109 49 75)(4 94 42 83 19 110 50 61)(5 95 43 84 20 111 51 62)(6 96 44 85 21 112 52 63)(7 97 45 86 22 113 53 64)(8 98 31 87 23 114 54 65)(9 99 32 88 24 115 55 66)(10 100 33 89 25 116 56 67)(11 101 34 90 26 117 57 68)(12 102 35 76 27 118 58 69)(13 103 36 77 28 119 59 70)(14 104 37 78 29 120 60 71)(15 105 38 79 30 106 46 72)
(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 76)(70 77)(71 78)(72 79)(73 80)(74 81)(75 82)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 106)
(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 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)```

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 6A 6B 8A 8B 8C 8D 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 3 4 4 5 5 6 6 8 8 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 2 2 2 2 2 2 4 4 4 60 60 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 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 C4 S3 D4 D5 D6 D10 D12 C3⋊D4 C4×S3 D15 D20 C5⋊D4 C4×D5 D30 D60 C15⋊7D4 C4×D15 C4.D4 C12.46D4 C20.46D4 M4(2)⋊D15 kernel M4(2)⋊D15 C60.7C4 C15×M4(2) C2×D60 C22×D15 C5×M4(2) C60 C3×M4(2) C2×C20 C2×C12 C20 C20 C2×C10 M4(2) C12 C12 C2×C6 C2×C4 C4 C4 C22 C15 C5 C3 C1 # reps 1 1 1 1 4 1 2 2 1 2 2 2 2 4 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of M4(2)⋊D15 in GL4(𝔽241) generated by

 0 0 240 1 51 240 239 51 83 178 1 0 124 181 1 0
,
 1 0 0 0 0 1 0 0 234 123 240 0 234 123 0 240
,
 225 110 0 0 195 30 0 0 110 0 94 110 177 131 131 161
,
 63 94 0 0 127 178 0 0 110 0 94 110 84 147 84 147
`G:=sub<GL(4,GF(241))| [0,51,83,124,0,240,178,181,240,239,1,1,1,51,0,0],[1,0,234,234,0,1,123,123,0,0,240,0,0,0,0,240],[225,195,110,177,110,30,0,131,0,0,94,131,0,0,110,161],[63,127,110,84,94,178,0,147,0,0,94,84,0,0,110,147] >;`

M4(2)⋊D15 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes D_{15}`
`% in TeX`

`G:=Group("M4(2):D15");`
`// GroupNames label`

`G:=SmallGroup(480,183);`
`// by ID`

`G=gap.SmallGroup(480,183);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽