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## G = D15⋊Q8order 240 = 24·3·5

### The semidirect product of D15 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D15⋊Q8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2 — D15⋊Q8
 Lower central C15 — C30 — D15⋊Q8
 Upper central C1 — C2 — C4

Generators and relations for D15⋊Q8
G = < a,b,c,d | a15=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 312 in 76 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C2×C4 [×3], Q8 [×4], D5 [×2], C10, Dic3 [×2], Dic3, C12, C12 [×2], D6, C15, C2×Q8, Dic5 [×2], Dic5, C20, C20 [×2], D10, Dic6, Dic6 [×2], C4×S3 [×3], C3×Q8, D15 [×2], C30, Dic10, Dic10 [×2], C4×D5 [×3], C5×Q8, S3×Q8, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60, D30, Q8×D5, D30.C2 [×2], C15⋊Q8 [×2], C3×Dic10, C5×Dic6, C4×D15, D15⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, D10 [×3], C22×S3, C22×D5, S3×Q8, S3×D5, Q8×D5, C2×S3×D5, D15⋊Q8

Smallest permutation representation of D15⋊Q8
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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(61 76)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(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)
(1 46 28 39)(2 47 29 40)(3 48 30 41)(4 49 16 42)(5 50 17 43)(6 51 18 44)(7 52 19 45)(8 53 20 31)(9 54 21 32)(10 55 22 33)(11 56 23 34)(12 57 24 35)(13 58 25 36)(14 59 26 37)(15 60 27 38)(61 98 89 112)(62 99 90 113)(63 100 76 114)(64 101 77 115)(65 102 78 116)(66 103 79 117)(67 104 80 118)(68 105 81 119)(69 91 82 120)(70 92 83 106)(71 93 84 107)(72 94 85 108)(73 95 86 109)(74 96 87 110)(75 97 88 111)
(1 83 28 70)(2 79 29 66)(3 90 30 62)(4 86 16 73)(5 82 17 69)(6 78 18 65)(7 89 19 61)(8 85 20 72)(9 81 21 68)(10 77 22 64)(11 88 23 75)(12 84 24 71)(13 80 25 67)(14 76 26 63)(15 87 27 74)(31 108 53 94)(32 119 54 105)(33 115 55 101)(34 111 56 97)(35 107 57 93)(36 118 58 104)(37 114 59 100)(38 110 60 96)(39 106 46 92)(40 117 47 103)(41 113 48 99)(42 109 49 95)(43 120 50 91)(44 116 51 102)(45 112 52 98)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6 10A 10B 12A 12B 12C 15A 15B 20A 20B 20C 20D 20E 20F 30A 30B 60A 60B 60C 60D order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 10 10 12 12 12 15 15 20 20 20 20 20 20 30 30 60 60 60 60 size 1 1 15 15 2 2 6 6 10 10 30 2 2 2 2 2 4 20 20 4 4 4 4 12 12 12 12 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + + + - + - + image C1 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D10 D10 S3×Q8 S3×D5 Q8×D5 C2×S3×D5 D15⋊Q8 kernel D15⋊Q8 D30.C2 C15⋊Q8 C3×Dic10 C5×Dic6 C4×D15 Dic10 D15 Dic6 Dic5 C20 Dic3 C12 C5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 2 2 1 4 2 1 2 2 2 4

Matrix representation of D15⋊Q8 in GL6(𝔽61)

 60 17 0 0 0 0 44 44 0 0 0 0 0 0 1 46 0 0 0 0 49 59 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 17 0 0 0 0 0 1 0 0 0 0 0 0 1 46 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 12 0 0 0 0 10 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 49 60 0 0 0 0 0 0 1 34 0 0 0 0 43 60

`G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,10,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,49,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,34,60] >;`

D15⋊Q8 in GAP, Magma, Sage, TeX

`D_{15}\rtimes Q_8`
`% in TeX`

`G:=Group("D15:Q8");`
`// GroupNames label`

`G:=SmallGroup(240,131);`
`// by ID`

`G=gap.SmallGroup(240,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^15=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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