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## G = C20.3S4order 480 = 25·3·5

### 3rd non-split extension by C20 of S4 acting via S4/A4=C2

Aliases: C20.3S4, Q8.5D30, SL2(𝔽3)⋊4D10, C4.3(C5⋊S4), C4.A41D5, C4○D42D15, Q8⋊D152C2, C10.24(C2×S4), C52(C4.3S4), (C5×Q8).12D6, (C5×SL2(𝔽3))⋊4C22, C2.10(C2×C5⋊S4), (C5×C4.A4)⋊1C2, (C5×C4○D4)⋊2S3, SmallGroup(480,1032)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.3S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15 — C20.3S4
 Lower central C5×SL2(𝔽3) — C20.3S4
 Upper central C1 — C2 — C4

Generators and relations for C20.3S4
G = < a,b,c,d,e | a20=d3=e2=1, b2=c2=a10, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a10b, dbd-1=a10bc, ebe=bc, dcd-1=b, ece=a10c, ede=d-1 >

Subgroups: 890 in 84 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, C12, D6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, SL2(𝔽3), D12, D15, C30, C8⋊C22, C52C8, D20, C2×C20, C5×D4, C5×Q8, C22×D5, GL2(𝔽3), C4.A4, C60, D30, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, C4.3S4, C5×SL2(𝔽3), D60, D4⋊D10, Q8⋊D15, C5×C4.A4, C20.3S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.3S4, C5⋊S4, C2×C5⋊S4, C20.3S4

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

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 6 8A 8B 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 2 3 4 4 5 5 6 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 60 8 2 6 2 2 8 60 60 2 2 12 12 8 8 8 8 8 8 2 2 2 2 12 12 8 8 8 8 8 ··· 8

41 irreducible representations

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

Matrix representation of C20.3S4 in GL4(𝔽241) generated by

 147 154 87 154 0 60 87 0 0 154 147 0 87 154 0 60
,
 1 0 239 0 0 0 240 1 1 0 240 0 1 240 240 0
,
 240 0 0 2 240 0 1 1 0 240 0 1 240 0 0 1
,
 1 0 0 0 0 0 0 1 1 240 0 0 1 0 240 0
,
 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 240
`G:=sub<GL(4,GF(241))| [147,0,0,87,154,60,154,154,87,87,147,0,154,0,0,60],[1,0,1,1,0,0,0,240,239,240,240,240,0,1,0,0],[240,240,0,240,0,0,240,0,0,1,0,0,2,1,1,1],[1,0,1,1,0,0,240,0,0,0,0,240,0,1,0,0],[1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,240] >;`

C20.3S4 in GAP, Magma, Sage, TeX

`C_{20}._3S_4`
`% in TeX`

`G:=Group("C20.3S4");`
`// GroupNames label`

`G:=SmallGroup(480,1032);`
`// by ID`

`G=gap.SmallGroup(480,1032);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1688,170,1347,4204,3168,172,2525,1909,285,124]);`
`// 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^-1,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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