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## G = D6.F5order 240 = 24·3·5

### The non-split extension by D6 of F5 acting via F5/D5=C2

Aliases: D6.F5, C151M4(2), Dic5.9D6, Dic15.2C4, C5⋊C81S3, C15⋊C83C2, C52(C8⋊S3), C2.6(S3×F5), C6.6(C2×F5), C10.6(C4×S3), C30.6(C2×C4), (S3×C10).2C4, C31(C22.F5), (S3×Dic5).3C2, (C3×Dic5).9C22, (C3×C5⋊C8)⋊3C2, SmallGroup(240,100)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D6.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — D6.F5
 Lower central C15 — C30 — D6.F5
 Upper central C1 — C2

Generators and relations for D6.F5
G = < a,b,c,d | a6=b2=c5=1, d4=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >

Character table of D6.F5

 class 1 2A 2B 3 4A 4B 4C 5 6 8A 8B 8C 8D 10A 10B 10C 12A 12B 15 24A 24B 24C 24D 30 size 1 1 6 2 5 5 30 4 2 10 10 30 30 4 12 12 10 10 8 10 10 10 10 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 1 1 -i i -i i 1 1 1 -1 -1 1 i -i -i i 1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 1 -i i i -i 1 -1 -1 -1 -1 1 i -i -i i 1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 i -i i -i 1 1 1 -1 -1 1 -i i i -i 1 linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 1 i -i -i i 1 -1 -1 -1 -1 1 -i i i -i 1 linear of order 4 ρ9 2 2 0 -1 2 2 0 2 -1 2 2 0 0 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 -1 2 2 0 2 -1 -2 -2 0 0 2 0 0 -1 -1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ11 2 -2 0 2 -2i 2i 0 2 -2 0 0 0 0 -2 0 0 -2i 2i 2 0 0 0 0 -2 complex lifted from M4(2) ρ12 2 -2 0 2 2i -2i 0 2 -2 0 0 0 0 -2 0 0 2i -2i 2 0 0 0 0 -2 complex lifted from M4(2) ρ13 2 2 0 -1 -2 -2 0 2 -1 -2i 2i 0 0 2 0 0 1 1 -1 -i i i -i -1 complex lifted from C4×S3 ρ14 2 2 0 -1 -2 -2 0 2 -1 2i -2i 0 0 2 0 0 1 1 -1 i -i -i i -1 complex lifted from C4×S3 ρ15 2 -2 0 -1 -2i 2i 0 2 1 0 0 0 0 -2 0 0 i -i -1 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 1 complex lifted from C8⋊S3 ρ16 2 -2 0 -1 2i -2i 0 2 1 0 0 0 0 -2 0 0 -i i -1 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 1 complex lifted from C8⋊S3 ρ17 2 -2 0 -1 2i -2i 0 2 1 0 0 0 0 -2 0 0 -i i -1 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 1 complex lifted from C8⋊S3 ρ18 2 -2 0 -1 -2i 2i 0 2 1 0 0 0 0 -2 0 0 i -i -1 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 1 complex lifted from C8⋊S3 ρ19 4 4 -4 4 0 0 0 -1 4 0 0 0 0 -1 1 1 0 0 -1 0 0 0 0 -1 orthogonal lifted from C2×F5 ρ20 4 4 4 4 0 0 0 -1 4 0 0 0 0 -1 -1 -1 0 0 -1 0 0 0 0 -1 orthogonal lifted from F5 ρ21 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 1 -√5 √5 0 0 -1 0 0 0 0 1 symplectic lifted from C22.F5, Schur index 2 ρ22 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 1 √5 -√5 0 0 -1 0 0 0 0 1 symplectic lifted from C22.F5, Schur index 2 ρ23 8 8 0 -4 0 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 1 0 0 0 0 1 orthogonal lifted from S3×F5 ρ24 8 -8 0 -4 0 0 0 -2 4 0 0 0 0 2 0 0 0 0 1 0 0 0 0 -1 symplectic faithful, Schur index 2

Smallest permutation representation of D6.F5
On 120 points
Generators in S120
```(1 15 50 5 11 54)(2 16 51 6 12 55)(3 9 52 7 13 56)(4 10 53 8 14 49)(17 110 29 21 106 25)(18 111 30 22 107 26)(19 112 31 23 108 27)(20 105 32 24 109 28)(33 75 113 37 79 117)(34 76 114 38 80 118)(35 77 115 39 73 119)(36 78 116 40 74 120)(41 62 103 45 58 99)(42 63 104 46 59 100)(43 64 97 47 60 101)(44 57 98 48 61 102)(65 91 87 69 95 83)(66 92 88 70 96 84)(67 93 81 71 89 85)(68 94 82 72 90 86)
(1 54)(2 51)(3 56)(4 53)(5 50)(6 55)(7 52)(8 49)(9 13)(11 15)(18 22)(20 24)(25 110)(26 107)(27 112)(28 109)(29 106)(30 111)(31 108)(32 105)(33 37)(35 39)(41 45)(43 47)(57 102)(58 99)(59 104)(60 101)(61 98)(62 103)(63 100)(64 97)(66 70)(68 72)(73 119)(74 116)(75 113)(76 118)(77 115)(78 120)(79 117)(80 114)(81 89)(82 94)(83 91)(84 96)(85 93)(86 90)(87 95)(88 92)
(1 99 88 113 28)(2 114 100 29 81)(3 30 115 82 101)(4 83 31 102 116)(5 103 84 117 32)(6 118 104 25 85)(7 26 119 86 97)(8 87 27 98 120)(9 22 39 72 43)(10 65 23 44 40)(11 45 66 33 24)(12 34 46 17 67)(13 18 35 68 47)(14 69 19 48 36)(15 41 70 37 20)(16 38 42 21 71)(49 95 112 61 78)(50 62 96 79 105)(51 80 63 106 89)(52 107 73 90 64)(53 91 108 57 74)(54 58 92 75 109)(55 76 59 110 93)(56 111 77 94 60)
(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)```

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

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

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

D6.F5 is a maximal subgroup of   D12.2F5  D12.F5  D15⋊M4(2)  C5⋊C8⋊D6  C5⋊C8.D6  S3×C22.F5  D15⋊C8⋊C2
D6.F5 is a maximal quotient of   C30.M4(2)  Dic5.22D12  Dic15⋊C8

Matrix representation of D6.F5 in GL6(𝔽241)

 240 240 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 240 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 234 123 1 0 0 0 234 123 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 52 240 0 0 0 0 53 240 0 0 0 0 51 51 190 190 0 0 1 0 51 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 1 0 0 190 1 239 51 0 0 205 139 240 0 0 0 141 139 240 0

`G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,234,234,0,0,0,240,123,123,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,53,51,1,0,0,240,240,51,0,0,0,0,0,190,51,0,0,0,0,190,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,190,205,141,0,0,0,1,139,139,0,0,240,239,240,240,0,0,1,51,0,0] >;`

D6.F5 in GAP, Magma, Sage, TeX

`D_6.F_5`
`% in TeX`

`G:=Group("D6.F5");`
`// GroupNames label`

`G:=SmallGroup(240,100);`
`// by ID`

`G=gap.SmallGroup(240,100);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,490,3461,1745]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^5=1,d^4=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;`
`// generators/relations`

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