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## G = C23.11D10order 160 = 25·5

### 1st non-split extension by C23 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C23.11D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.11D10
 Lower central C5 — C10 — C23.11D10
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C23.11D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Subgroups: 184 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C23.11D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C4×D5, C22×D5, C2×C4×D5, D42D5, C23.11D10

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

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 C4×D5 D4⋊2D5 kernel C23.11D10 C4×Dic5 C10.D4 C23.D5 C5×C22⋊C4 C22×Dic5 C2×Dic5 C22⋊C4 C10 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 8 2 4 4 2 8 4

Matrix representation of C23.11D10 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 0 6 0 0 0 34 7
,
 32 0 0 0 0 0 0 32 0 0 0 9 0 0 0 0 0 0 2 36 0 0 0 17 39

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,34,0,0,0,6,7],[32,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,2,17,0,0,0,36,39] >;`

C23.11D10 in GAP, Magma, Sage, TeX

`C_2^3._{11}D_{10}`
`% in TeX`

`G:=Group("C2^3.11D10");`
`// GroupNames label`

`G:=SmallGroup(160,98);`
`// by ID`

`G=gap.SmallGroup(160,98);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,188,50,4613]);`
`// Polycyclic`

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

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