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## G = C12.F5order 240 = 24·3·5

### 1st non-split extension by C12 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C12.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊C8 — C12.F5
 Lower central C15 — C30 — C12.F5
 Upper central C1 — C2 — C4

Generators and relations for C12.F5
G = < a,b,c | a12=b5=1, c4=a6, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C12.F5

 class 1 2A 2B 3 4A 4B 4C 5 6A 6B 6C 8A 8B 8C 8D 10 12A 12B 12C 12D 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D size 1 1 10 2 2 5 5 4 2 10 10 30 30 30 30 4 2 2 10 10 4 4 4 4 4 4 4 4 4 4 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 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 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 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 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 1 1 -1 -1 1 1 -1 -1 i i -i -i 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 1 1 1 1 -i i i -i 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 1 1 i -i -i i 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 1 1 -1 -1 1 1 -1 -1 -i -i i i 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 -1 2 2 2 2 -1 -1 -1 0 0 0 0 2 -1 -1 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -1 -2 2 2 2 -1 1 1 0 0 0 0 2 1 1 -1 -1 -1 -1 -2 -2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 -1 2 -2 -2 2 -1 1 1 0 0 0 0 2 -1 -1 1 1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 2 -1 -2 -2 -2 2 -1 -1 -1 0 0 0 0 2 1 1 1 1 -1 -1 -2 -2 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 0 2 0 -2i 2i 2 -2 0 0 0 0 0 0 -2 0 0 -2i 2i 2 2 0 0 -2 -2 0 0 0 0 complex lifted from M4(2) ρ14 2 -2 0 2 0 2i -2i 2 -2 0 0 0 0 0 0 -2 0 0 2i -2i 2 2 0 0 -2 -2 0 0 0 0 complex lifted from M4(2) ρ15 2 -2 0 -1 0 -2i 2i 2 1 -√-3 √-3 0 0 0 0 -2 -√3 √3 i -i -1 -1 0 0 1 1 √3 -√3 -√3 √3 complex lifted from C4.Dic3 ρ16 2 -2 0 -1 0 2i -2i 2 1 -√-3 √-3 0 0 0 0 -2 √3 -√3 -i i -1 -1 0 0 1 1 -√3 √3 √3 -√3 complex lifted from C4.Dic3 ρ17 2 -2 0 -1 0 2i -2i 2 1 √-3 -√-3 0 0 0 0 -2 -√3 √3 -i i -1 -1 0 0 1 1 √3 -√3 -√3 √3 complex lifted from C4.Dic3 ρ18 2 -2 0 -1 0 -2i 2i 2 1 √-3 -√-3 0 0 0 0 -2 √3 -√3 i -i -1 -1 0 0 1 1 -√3 √3 √3 -√3 complex lifted from C4.Dic3 ρ19 4 4 0 4 4 0 0 -1 4 0 0 0 0 0 0 -1 4 4 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 0 4 -4 0 0 -1 4 0 0 0 0 0 0 -1 -4 -4 0 0 -1 -1 1 1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ21 4 4 0 -2 -4 0 0 -1 -2 0 0 0 0 0 0 -1 2 2 0 0 1+√-15/2 1-√-15/2 1 1 1-√-15/2 1+√-15/2 -1-√-15/2 -1+√-15/2 -1-√-15/2 -1+√-15/2 complex lifted from C2×C3⋊F5 ρ22 4 4 0 -2 4 0 0 -1 -2 0 0 0 0 0 0 -1 -2 -2 0 0 1-√-15/2 1+√-15/2 -1 -1 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ23 4 4 0 -2 4 0 0 -1 -2 0 0 0 0 0 0 -1 -2 -2 0 0 1+√-15/2 1-√-15/2 -1 -1 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ24 4 4 0 -2 -4 0 0 -1 -2 0 0 0 0 0 0 -1 2 2 0 0 1-√-15/2 1+√-15/2 1 1 1+√-15/2 1-√-15/2 -1+√-15/2 -1-√-15/2 -1+√-15/2 -1-√-15/2 complex lifted from C2×C3⋊F5 ρ25 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 0 0 1 0 0 0 0 -1 -1 √-5 -√-5 1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4.F5 ρ26 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -√-5 √-5 1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4.F5 ρ27 4 -4 0 -2 0 0 0 -1 2 0 0 0 0 0 0 1 2√3 -2√3 0 0 1+√-15/2 1-√-15/2 √-5 -√-5 -1+√-15/2 -1-√-15/2 -ζ4ζ3+ζ4ζ53+ζ4ζ52 -ζ43ζ3+ζ43ζ54+ζ43ζ5 -ζ4ζ32+ζ4ζ54+ζ4ζ5 -ζ43ζ32+ζ43ζ53+ζ43ζ52 complex faithful ρ28 4 -4 0 -2 0 0 0 -1 2 0 0 0 0 0 0 1 -2√3 2√3 0 0 1-√-15/2 1+√-15/2 √-5 -√-5 -1-√-15/2 -1+√-15/2 -ζ43ζ3+ζ43ζ54+ζ43ζ5 -ζ4ζ3+ζ4ζ53+ζ4ζ52 -ζ43ζ32+ζ43ζ53+ζ43ζ52 -ζ4ζ32+ζ4ζ54+ζ4ζ5 complex faithful ρ29 4 -4 0 -2 0 0 0 -1 2 0 0 0 0 0 0 1 -2√3 2√3 0 0 1+√-15/2 1-√-15/2 -√-5 √-5 -1+√-15/2 -1-√-15/2 -ζ4ζ32+ζ4ζ54+ζ4ζ5 -ζ43ζ32+ζ43ζ53+ζ43ζ52 -ζ4ζ3+ζ4ζ53+ζ4ζ52 -ζ43ζ3+ζ43ζ54+ζ43ζ5 complex faithful ρ30 4 -4 0 -2 0 0 0 -1 2 0 0 0 0 0 0 1 2√3 -2√3 0 0 1-√-15/2 1+√-15/2 -√-5 √-5 -1-√-15/2 -1+√-15/2 -ζ43ζ32+ζ43ζ53+ζ43ζ52 -ζ4ζ32+ζ4ζ54+ζ4ζ5 -ζ43ζ3+ζ43ζ54+ζ43ζ5 -ζ4ζ3+ζ4ζ53+ζ4ζ52 complex faithful

Smallest permutation representation of C12.F5
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 83 31 105 17)(2 84 32 106 18)(3 73 33 107 19)(4 74 34 108 20)(5 75 35 97 21)(6 76 36 98 22)(7 77 25 99 23)(8 78 26 100 24)(9 79 27 101 13)(10 80 28 102 14)(11 81 29 103 15)(12 82 30 104 16)(37 50 62 93 118)(38 51 63 94 119)(39 52 64 95 120)(40 53 65 96 109)(41 54 66 85 110)(42 55 67 86 111)(43 56 68 87 112)(44 57 69 88 113)(45 58 70 89 114)(46 59 71 90 115)(47 60 72 91 116)(48 49 61 92 117)
(1 42 10 45 7 48 4 39)(2 41 11 44 8 47 5 38)(3 40 12 43 9 46 6 37)(13 90 76 62 19 96 82 68)(14 89 77 61 20 95 83 67)(15 88 78 72 21 94 84 66)(16 87 79 71 22 93 73 65)(17 86 80 70 23 92 74 64)(18 85 81 69 24 91 75 63)(25 117 108 52 31 111 102 58)(26 116 97 51 32 110 103 57)(27 115 98 50 33 109 104 56)(28 114 99 49 34 120 105 55)(29 113 100 60 35 119 106 54)(30 112 101 59 36 118 107 53)```

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

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

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

C12.F5 is a maximal subgroup of
D122F5  D605C4  D10.Dic6  D10.2Dic6  C40.Dic3  C24.1F5  Dic10⋊Dic3  D202Dic3  D12.2F5  S3×C4.F5  D60.C4  D15⋊M4(2)  C60.59(C2×C4)  Dic10.Dic3  D20.Dic3
C12.F5 is a maximal quotient of
C60⋊C8  C30.11C42  C30.7M4(2)

Matrix representation of C12.F5 in GL6(𝔽241)

 4 0 0 0 0 0 238 181 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 15 79 0 0 0 0 36 226 0 0 0 0 0 0 24 0 191 191 0 0 191 191 0 24 0 0 50 74 50 0 0 0 217 167 167 217

`G:=sub<GL(6,GF(241))| [4,238,0,0,0,0,0,181,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[15,36,0,0,0,0,79,226,0,0,0,0,0,0,24,191,50,217,0,0,0,191,74,167,0,0,191,0,50,167,0,0,191,24,0,217] >;`

C12.F5 in GAP, Magma, Sage, TeX

`C_{12}.F_5`
`% in TeX`

`G:=Group("C12.F5");`
`// GroupNames label`

`G:=SmallGroup(240,119);`
`// by ID`

`G=gap.SmallGroup(240,119);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,50,964,5189,1745]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^5=1,c^4=a^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;`
`// generators/relations`

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