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## G = C20.23D4order 160 = 25·5

### 23rd non-split extension by C20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.23D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C20.23D4
 Lower central C5 — C2×C10 — C20.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C20.23D4
G = < a,b,c | a20=b4=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >

Subgroups: 272 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4.4D4, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C20.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C5⋊D4 [×2], C22×D5, Q82D5 [×2], C2×C5⋊D4, C20.23D4

Smallest permutation representation of C20.23D4
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 59 62 33)(2 48 63 22)(3 57 64 31)(4 46 65 40)(5 55 66 29)(6 44 67 38)(7 53 68 27)(8 42 69 36)(9 51 70 25)(10 60 71 34)(11 49 72 23)(12 58 73 32)(13 47 74 21)(14 56 75 30)(15 45 76 39)(16 54 77 28)(17 43 78 37)(18 52 79 26)(19 41 80 35)(20 50 61 24)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 C4○D4 D10 C5⋊D4 Q8⋊2D5 kernel C20.23D4 C4×Dic5 D10⋊C4 C2×D20 Q8×C10 C20 C2×Q8 C10 C2×C4 C4 C2 # reps 1 1 4 1 1 2 2 4 6 8 4

Matrix representation of C20.23D4 in GL4(𝔽41) generated by

 0 1 0 0 40 0 0 0 0 0 35 40 0 0 36 40
,
 9 0 0 0 0 9 0 0 0 0 21 17 0 0 15 20
,
 1 0 0 0 0 40 0 0 0 0 6 1 0 0 6 35
`G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,35,36,0,0,40,40],[9,0,0,0,0,9,0,0,0,0,21,15,0,0,17,20],[1,0,0,0,0,40,0,0,0,0,6,6,0,0,1,35] >;`

C20.23D4 in GAP, Magma, Sage, TeX

`C_{20}._{23}D_4`
`% in TeX`

`G:=Group("C20.23D4");`
`// GroupNames label`

`G:=SmallGroup(160,168);`
`// by ID`

`G=gap.SmallGroup(160,168);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,188,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;`
`// generators/relations`

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