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## G = C20.C23order 160 = 25·5

### 15th non-split extension by C20 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.C23
 Chief series C1 — C5 — C10 — C20 — D20 — C4○D20 — C20.C23
 Lower central C5 — C10 — C20 — C20.C23
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C20.C23
G = < a,b,c,d | a20=b2=1, c2=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a5b, dcd-1=a10c >

Subgroups: 176 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C8.C22, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, C20.C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, C5⋊D4, C22×D5, C2×C5⋊D4, C20.C23

Smallest permutation representation of C20.C23
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)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 56 11 46)(2 57 12 47)(3 58 13 48)(4 59 14 49)(5 60 15 50)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 36 11 26)(2 27 12 37)(3 38 13 28)(4 29 14 39)(5 40 15 30)(6 31 16 21)(7 22 17 32)(8 33 18 23)(9 24 19 34)(10 35 20 25)(41 61 51 71)(42 72 52 62)(43 63 53 73)(44 74 54 64)(45 65 55 75)(46 76 56 66)(47 67 57 77)(48 78 58 68)(49 69 59 79)(50 80 60 70)```

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

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

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

31 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 5 5 8 8 10 ··· 10 20 ··· 20 size 1 1 2 20 2 2 4 4 20 2 2 20 20 2 ··· 2 4 ··· 4

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8.C22 C20.C23 kernel C20.C23 C4.Dic5 Q8⋊D5 C5⋊Q16 C4○D20 Q8×C10 C20 C2×C10 C2×Q8 C2×C4 Q8 C4 C22 C5 C1 # reps 1 1 2 2 1 1 1 1 2 2 4 4 4 1 4

Matrix representation of C20.C23 in GL4(𝔽41) generated by

 0 40 28 13 1 6 24 28 14 32 35 1 11 14 40 0
,
 6 35 0 0 40 35 0 0 11 14 40 0 14 32 35 1
,
 18 6 8 26 35 23 8 8 32 19 18 35 9 32 6 23
,
 4 0 6 29 0 4 17 6 17 34 37 0 27 17 0 37
`G:=sub<GL(4,GF(41))| [0,1,14,11,40,6,32,14,28,24,35,40,13,28,1,0],[6,40,11,14,35,35,14,32,0,0,40,35,0,0,0,1],[18,35,32,9,6,23,19,32,8,8,18,6,26,8,35,23],[4,0,17,27,0,4,34,17,6,17,37,0,29,6,0,37] >;`

C20.C23 in GAP, Magma, Sage, TeX

`C_{20}.C_2^3`
`% in TeX`

`G:=Group("C20.C2^3");`
`// GroupNames label`

`G:=SmallGroup(160,163);`
`// by ID`

`G=gap.SmallGroup(160,163);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,86,579,159,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^10*c>;`
`// generators/relations`

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