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## G = C20.6S4order 480 = 25·3·5

### 6th non-split extension by C20 of S4 acting via S4/A4=C2

Aliases: C20.6S4, Q8.4D30, SL2(𝔽3).10D10, C4.6(C5⋊S4), C4.A42D5, C4○D41D15, Q8⋊D157C2, C10.23(C2×S4), C53(C4.6S4), Q8.D157C2, (C5×Q8).11D6, (C5×SL2(𝔽3)).10C22, C2.9(C2×C5⋊S4), (C5×C4.A4)⋊2C2, (C5×C4○D4)⋊1S3, SmallGroup(480,1031)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.6S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15 — C20.6S4
 Lower central C5×SL2(𝔽3) — C20.6S4
 Upper central C1 — C4

Generators and relations for C20.6S4
G = < a,b,c,d,e | a20=d3=e2=1, b2=c2=a10, ab=ba, ac=ca, ad=da, eae=a9, cbc-1=a10b, dbd-1=a10bc, ebe=bc, dcd-1=b, ece=a10c, ede=d-1 >

Subgroups: 626 in 78 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, SL2(𝔽3), C4×S3, D15, C30, C4○D8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, Dic15, C60, D30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C4.6S4, C5×SL2(𝔽3), C4×D15, D4.8D10, Q8.D15, Q8⋊D15, C5×C4.A4, C20.6S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.6S4, C5⋊S4, C2×C5⋊S4, C20.6S4

Smallest permutation representation of C20.6S4
On 80 points
Generators in S80
```(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)
(1 48 11 58)(2 49 12 59)(3 50 13 60)(4 51 14 41)(5 52 15 42)(6 53 16 43)(7 54 17 44)(8 55 18 45)(9 56 19 46)(10 57 20 47)(21 64 31 74)(22 65 32 75)(23 66 33 76)(24 67 34 77)(25 68 35 78)(26 69 36 79)(27 70 37 80)(28 71 38 61)(29 72 39 62)(30 73 40 63)
(1 74 11 64)(2 75 12 65)(3 76 13 66)(4 77 14 67)(5 78 15 68)(6 79 16 69)(7 80 17 70)(8 61 18 71)(9 62 19 72)(10 63 20 73)(21 48 31 58)(22 49 32 59)(23 50 33 60)(24 51 34 41)(25 52 35 42)(26 53 36 43)(27 54 37 44)(28 55 38 45)(29 56 39 46)(30 57 40 47)
(21 48 74)(22 49 75)(23 50 76)(24 51 77)(25 52 78)(26 53 79)(27 54 80)(28 55 61)(29 56 62)(30 57 63)(31 58 64)(32 59 65)(33 60 66)(34 41 67)(35 42 68)(36 43 69)(37 44 70)(38 45 71)(39 46 72)(40 47 73)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 48)(22 57)(23 46)(24 55)(25 44)(26 53)(27 42)(28 51)(29 60)(30 49)(31 58)(32 47)(33 56)(34 45)(35 54)(36 43)(37 52)(38 41)(39 50)(40 59)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)```

`G:=sub<Sym(80)| (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), (1,48,11,58)(2,49,12,59)(3,50,13,60)(4,51,14,41)(5,52,15,42)(6,53,16,43)(7,54,17,44)(8,55,18,45)(9,56,19,46)(10,57,20,47)(21,64,31,74)(22,65,32,75)(23,66,33,76)(24,67,34,77)(25,68,35,78)(26,69,36,79)(27,70,37,80)(28,71,38,61)(29,72,39,62)(30,73,40,63), (1,74,11,64)(2,75,12,65)(3,76,13,66)(4,77,14,67)(5,78,15,68)(6,79,16,69)(7,80,17,70)(8,61,18,71)(9,62,19,72)(10,63,20,73)(21,48,31,58)(22,49,32,59)(23,50,33,60)(24,51,34,41)(25,52,35,42)(26,53,36,43)(27,54,37,44)(28,55,38,45)(29,56,39,46)(30,57,40,47), (21,48,74)(22,49,75)(23,50,76)(24,51,77)(25,52,78)(26,53,79)(27,54,80)(28,55,61)(29,56,62)(30,57,63)(31,58,64)(32,59,65)(33,60,66)(34,41,67)(35,42,68)(36,43,69)(37,44,70)(38,45,71)(39,46,72)(40,47,73), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,48)(22,57)(23,46)(24,55)(25,44)(26,53)(27,42)(28,51)(29,60)(30,49)(31,58)(32,47)(33,56)(34,45)(35,54)(36,43)(37,52)(38,41)(39,50)(40,59)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)>;`

`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), (1,48,11,58)(2,49,12,59)(3,50,13,60)(4,51,14,41)(5,52,15,42)(6,53,16,43)(7,54,17,44)(8,55,18,45)(9,56,19,46)(10,57,20,47)(21,64,31,74)(22,65,32,75)(23,66,33,76)(24,67,34,77)(25,68,35,78)(26,69,36,79)(27,70,37,80)(28,71,38,61)(29,72,39,62)(30,73,40,63), (1,74,11,64)(2,75,12,65)(3,76,13,66)(4,77,14,67)(5,78,15,68)(6,79,16,69)(7,80,17,70)(8,61,18,71)(9,62,19,72)(10,63,20,73)(21,48,31,58)(22,49,32,59)(23,50,33,60)(24,51,34,41)(25,52,35,42)(26,53,36,43)(27,54,37,44)(28,55,38,45)(29,56,39,46)(30,57,40,47), (21,48,74)(22,49,75)(23,50,76)(24,51,77)(25,52,78)(26,53,79)(27,54,80)(28,55,61)(29,56,62)(30,57,63)(31,58,64)(32,59,65)(33,60,66)(34,41,67)(35,42,68)(36,43,69)(37,44,70)(38,45,71)(39,46,72)(40,47,73), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,48)(22,57)(23,46)(24,55)(25,44)(26,53)(27,42)(28,51)(29,60)(30,49)(31,58)(32,47)(33,56)(34,45)(35,54)(36,43)(37,52)(38,41)(39,50)(40,59)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76) );`

`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)], [(1,48,11,58),(2,49,12,59),(3,50,13,60),(4,51,14,41),(5,52,15,42),(6,53,16,43),(7,54,17,44),(8,55,18,45),(9,56,19,46),(10,57,20,47),(21,64,31,74),(22,65,32,75),(23,66,33,76),(24,67,34,77),(25,68,35,78),(26,69,36,79),(27,70,37,80),(28,71,38,61),(29,72,39,62),(30,73,40,63)], [(1,74,11,64),(2,75,12,65),(3,76,13,66),(4,77,14,67),(5,78,15,68),(6,79,16,69),(7,80,17,70),(8,61,18,71),(9,62,19,72),(10,63,20,73),(21,48,31,58),(22,49,32,59),(23,50,33,60),(24,51,34,41),(25,52,35,42),(26,53,36,43),(27,54,37,44),(28,55,38,45),(29,56,39,46),(30,57,40,47)], [(21,48,74),(22,49,75),(23,50,76),(24,51,77),(25,52,78),(26,53,79),(27,54,80),(28,55,61),(29,56,62),(30,57,63),(31,58,64),(32,59,65),(33,60,66),(34,41,67),(35,42,68),(36,43,69),(37,44,70),(38,45,71),(39,46,72),(40,47,73)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,48),(22,57),(23,46),(24,55),(25,44),(26,53),(27,42),(28,51),(29,60),(30,49),(31,58),(32,47),(33,56),(34,45),(35,54),(36,43),(37,52),(38,41),(39,50),(40,59),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76)]])`

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 3 3 4 4 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 D10 D15 D30 C4.6S4 S4 C2×S4 C4.6S4 C20.6S4 C5⋊S4 C2×C5⋊S4 kernel C20.6S4 Q8.D15 Q8⋊D15 C5×C4.A4 C5×C4○D4 C4.A4 C5×Q8 SL2(𝔽3) C4○D4 Q8 C5 C20 C10 C5 C1 C4 C2 # reps 1 1 1 1 1 2 1 2 4 4 4 2 2 2 12 2 2

Matrix representation of C20.6S4 in GL4(𝔽241) generated by

 219 233 0 0 8 211 0 0 0 0 177 0 0 0 0 177
,
 1 0 0 0 0 1 0 0 0 0 13 12 0 0 26 228
,
 1 0 0 0 0 1 0 0 0 0 216 12 0 0 229 25
,
 0 240 0 0 1 240 0 0 0 0 240 1 0 0 240 0
,
 1 240 0 0 0 240 0 0 0 0 0 240 0 0 240 0
`G:=sub<GL(4,GF(241))| [219,8,0,0,233,211,0,0,0,0,177,0,0,0,0,177],[1,0,0,0,0,1,0,0,0,0,13,26,0,0,12,228],[1,0,0,0,0,1,0,0,0,0,216,229,0,0,12,25],[0,1,0,0,240,240,0,0,0,0,240,240,0,0,1,0],[1,0,0,0,240,240,0,0,0,0,0,240,0,0,240,0] >;`

C20.6S4 in GAP, Magma, Sage, TeX

`C_{20}._6S_4`
`% in TeX`

`G:=Group("C20.6S4");`
`// GroupNames label`

`G:=SmallGroup(480,1031);`
`// by ID`

`G=gap.SmallGroup(480,1031);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1688,170,1347,4204,3168,172,2525,1909,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=d^3=e^2=1,b^2=c^2=a^10,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^9,c*b*c^-1=a^10*b,d*b*d^-1=a^10*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^10*c,e*d*e=d^-1>;`
`// generators/relations`

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