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G = D15⋊4M4(2)  order 480 = 25·3·5

The semidirect product of D15 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D15⋊4M4(2)
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D15⋊2C8 — D15⋊4M4(2)
 Lower central C15 — C30 — D15⋊4M4(2)
 Upper central C1 — C4 — C2×C4

Generators and relations for D154M4(2)
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd=c5 >

Subgroups: 668 in 136 conjugacy classes, 54 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×3], C6, C6, C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×3], C10, C10, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], D15, C30, C30, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8 [×2], C3×C52C8 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, D5×M4(2), D152C8 [×2], D30.5C4 [×2], C3×C4.Dic5, C5×C4.Dic3, C2×C4×D15, D154M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), D30.C2 [×2], C2×S3×D5, D5×M4(2), C2×D30.C2, D154M4(2)

Smallest permutation representation of D154M4(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(76 77)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 101)(92 100)(93 99)(94 98)(95 97)(102 105)(103 104)(106 108)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 115 43 68 28 104 46 77)(2 111 44 64 29 100 47 88)(3 107 45 75 30 96 48 84)(4 118 31 71 16 92 49 80)(5 114 32 67 17 103 50 76)(6 110 33 63 18 99 51 87)(7 106 34 74 19 95 52 83)(8 117 35 70 20 91 53 79)(9 113 36 66 21 102 54 90)(10 109 37 62 22 98 55 86)(11 120 38 73 23 94 56 82)(12 116 39 69 24 105 57 78)(13 112 40 65 25 101 58 89)(14 108 41 61 26 97 59 85)(15 119 42 72 27 93 60 81)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 46)(44 47)(45 48)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 12A 12B 12C 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A ··· 30F 40A ··· 40H 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 15 15 20 20 20 20 20 20 24 24 24 24 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 15 15 30 2 1 1 2 15 15 30 2 2 2 4 6 6 6 6 10 10 10 10 2 2 4 4 2 2 4 4 4 2 2 2 2 4 4 20 20 20 20 4 ··· 4 12 ··· 12 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D5 D6 D6 M4(2) D10 D10 C4×S3 C4×S3 C4×D5 C4×D5 S3×D5 S3×M4(2) D30.C2 C2×S3×D5 D30.C2 D5×M4(2) D15⋊4M4(2) kernel D15⋊4M4(2) D15⋊2C8 D30.5C4 C3×C4.Dic5 C5×C4.Dic3 C2×C4×D15 C4×D15 C2×Dic15 C22×D15 C4.Dic5 C4.Dic3 C5⋊2C8 C2×C20 D15 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 2 1 4 4 2 2 2 4 4 2 2 2 2 2 4 8

Matrix representation of D154M4(2) in GL6(𝔽241)

 0 51 0 0 0 0 189 189 0 0 0 0 0 0 240 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 178 84 0 0 0 0 79 63 0 0 0 0 0 0 240 0 0 0 0 0 240 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 0 64 0 0 0 0 0 0 0 177 0 0 0 0 177 0 0 0 0 0 0 0 0 240 0 0 0 0 64 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [0,189,0,0,0,0,51,189,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[178,79,0,0,0,0,84,63,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;`

D154M4(2) in GAP, Magma, Sage, TeX

`D_{15}\rtimes_4M_4(2)`
`% in TeX`

`G:=Group("D15:4M4(2)");`
`// GroupNames label`

`G:=SmallGroup(480,368);`
`// by ID`

`G=gap.SmallGroup(480,368);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,80,1356,18822]);`
`// 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^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

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