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## G = C23.23D10order 160 = 25·5

### 4th non-split extension by C23 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23.23D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C23.23D10
 Lower central C5 — C2×C10 — C23.23D10
 Upper central C1 — C22 — C22×C4

Generators and relations for C23.23D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd9 >

Subgroups: 240 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.23D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C5⋊D4, C22×D5, C4○D20, C2×C5⋊D4, C23.23D10

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

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C5⋊D4 C4○D20 kernel C23.23D10 C10.D4 D10⋊C4 C23.D5 C2×C5⋊D4 C22×C20 C2×C10 C22×C4 C10 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 2 2 4 4 2 8 16

Matrix representation of C23.23D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 24 1 0 0 40 17
,
 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
,
 35 26 0 0 38 20 0 0 0 0 19 19 0 0 22 9
,
 21 26 0 0 24 20 0 0 0 0 19 19 0 0 9 22
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,24,40,0,0,1,17],[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],[35,38,0,0,26,20,0,0,0,0,19,22,0,0,19,9],[21,24,0,0,26,20,0,0,0,0,19,9,0,0,19,22] >;`

C23.23D10 in GAP, Magma, Sage, TeX

`C_2^3._{23}D_{10}`
`% in TeX`

`G:=Group("C2^3.23D10");`
`// GroupNames label`

`G:=SmallGroup(160,150);`
`// by ID`

`G=gap.SmallGroup(160,150);`
`# by ID`

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

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

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