Copied to
clipboard

## G = C23.D10order 160 = 25·5

### 3rd non-split extension by C23 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 160 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C422C2, C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C23.D10
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4 [×3], D10 [×3], C422C2, C22×D5, C4○D20, D42D5 [×2], C23.D10

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 1 1 4 2 2 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 1 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 C4○D20 D4⋊2D5 kernel C23.D10 C4×Dic5 C10.D4 C4⋊Dic5 C23.D5 C5×C22⋊C4 C22⋊C4 C10 C2×C4 C23 C2 C2 # reps 1 1 2 1 2 1 2 6 4 2 8 4

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

 1 0 0 0 6 40 0 0 0 0 1 0 0 0 25 40
,
 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
,
 8 0 0 0 9 5 0 0 0 0 40 5 0 0 0 1
,
 30 31 0 0 4 11 0 0 0 0 32 0 0 0 0 32
`G:=sub<GL(4,GF(41))| [1,6,0,0,0,40,0,0,0,0,1,25,0,0,0,40],[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],[8,9,0,0,0,5,0,0,0,0,40,0,0,0,5,1],[30,4,0,0,31,11,0,0,0,0,32,0,0,0,0,32] >;`

C23.D10 in GAP, Magma, Sage, TeX

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

`G:=Group("C2^3.D10");`
`// GroupNames label`

`G:=SmallGroup(160,100);`
`// by ID`

`G=gap.SmallGroup(160,100);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,188,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-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`

׿
×
𝔽