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## G = C23.18D10order 160 = 25·5

### 8th non-split extension by C23 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23.18D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.18D10
 Lower central C5 — C2×C10 — C23.18D10
 Upper central C1 — C22 — C2×D4

Generators and relations for C23.18D10
G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 208 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×4], C20, C2×C10, C2×C10 [×2], C2×C10 [×5], C22.D4, C2×Dic5 [×4], C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C10.D4 [×2], C23.D5, C23.D5 [×2], C22×Dic5, D4×C10, C23.18D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C5⋊D4 [×2], C22×D5, D42D5 [×2], C2×C5⋊D4, C23.18D10

Smallest permutation representation of C23.18D10
On 80 points
Generators in S80
```(1 21)(2 27)(3 23)(4 29)(5 25)(6 24)(7 30)(8 26)(9 22)(10 28)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 54)(42 72)(43 56)(44 74)(45 58)(46 76)(47 60)(48 78)(49 52)(50 80)(51 66)(53 68)(55 70)(57 62)(59 64)(61 73)(63 75)(65 77)(67 79)(69 71)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 46)(42 47)(43 48)(44 49)(45 50)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 71)(60 72)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(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 41 18 64)(2 45 19 68)(3 49 20 62)(4 43 16 66)(5 47 17 70)(6 61 11 48)(7 65 12 42)(8 69 13 46)(9 63 14 50)(10 67 15 44)(21 71 38 76)(22 58 39 53)(23 79 40 74)(24 56 31 51)(25 77 32 72)(26 54 33 59)(27 75 34 80)(28 52 35 57)(29 73 36 78)(30 60 37 55)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C5⋊D4 D4⋊2D5 kernel C23.18D10 C10.D4 C23.D5 C22×Dic5 D4×C10 C2×C10 C2×D4 C10 C2×C4 C23 C22 C2 # reps 1 2 3 1 1 2 2 4 2 4 8 4

Matrix representation of C23.18D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 22 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 16 0 0 0 19 23 0 0 0 0 1 0 0 0 15 40
,
 18 39 0 0 19 23 0 0 0 0 9 0 0 0 12 32
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,22,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,19,0,0,0,23,0,0,0,0,1,15,0,0,0,40],[18,19,0,0,39,23,0,0,0,0,9,12,0,0,0,32] >;`

C23.18D10 in GAP, Magma, Sage, TeX

`C_2^3._{18}D_{10}`
`% in TeX`

`G:=Group("C2^3.18D10");`
`// GroupNames label`

`G:=SmallGroup(160,156);`
`// by ID`

`G=gap.SmallGroup(160,156);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,4613]);`
`// Polycyclic`

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

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