Copied to
clipboard

## G = C42⋊23D10order 320 = 26·5

### 23rd semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊23D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C42⋊23D10
 Lower central C5 — C2×C10 — C42⋊23D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4223D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >

Subgroups: 1134 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C5, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic5 [×4], C20 [×6], D10 [×2], D10 [×15], C2×C10, C2×C10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2, C422C2, Dic10, C4×D5 [×4], D20 [×7], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×6], C22×D5 [×4], C22×D5 [×4], C22×C10, C22.32C24, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×10], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×5], C2×C5⋊D4 [×2], C23×D5, C4×D20, C4.D20, D5×C22⋊C4, C22⋊D20 [×2], D10.12D4, D10⋊D4, Dic5.5D4, D208C4, D10.13D4, C4⋊D20 [×2], D10⋊Q8, C4⋊C4⋊D5, C5×C422C2, C4223D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.32C24, C23×D5, D5×C4○D4, D48D10 [×2], C4223D10

Smallest permutation representation of C4223D10
On 80 points
Generators in S80
```(1 65 8 44)(2 61 9 50)(3 67 10 46)(4 63 6 42)(5 69 7 48)(11 68 16 47)(12 64 17 43)(13 70 18 49)(14 66 19 45)(15 62 20 41)(21 78 26 57)(22 53 27 74)(23 80 28 59)(24 55 29 76)(25 72 30 51)(31 60 36 71)(32 77 37 56)(33 52 38 73)(34 79 39 58)(35 54 40 75)
(1 38 18 21)(2 34 19 27)(3 40 20 23)(4 36 16 29)(5 32 17 25)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(41 80 67 75)(42 60 68 55)(43 72 69 77)(44 52 70 57)(45 74 61 79)(46 54 62 59)(47 76 63 71)(48 56 64 51)(49 78 65 73)(50 58 66 53)
(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 7)(2 6)(3 10)(4 9)(5 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 70)(49 69)(50 68)(51 57)(52 56)(53 55)(58 60)(71 79)(72 78)(73 77)(74 76)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G 10H 20A ··· 20L 20M ··· 20R order 1 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 4 4 4 4 5 5 10 ··· 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 10 10 20 20 20 2 2 4 ··· 4 10 10 20 20 20 2 2 2 ··· 2 8 8 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 D5×C4○D4 D4⋊8D10 kernel C42⋊23D10 C4×D20 C4.D20 D5×C22⋊C4 C22⋊D20 D10.12D4 D10⋊D4 Dic5.5D4 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊Q8 C4⋊C4⋊D5 C5×C42⋊2C2 C42⋊2C2 D10 C42 C22⋊C4 C4⋊C4 C10 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 4 2 6 6 2 4 8

Matrix representation of C4223D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 11 9 0 0 0 0 32 30 0 0
,
 1 37 0 0 0 0 21 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 21 40 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 34 34 0 0 0 0 7 1
,
 1 0 0 0 0 0 21 40 0 0 0 0 0 0 34 34 0 0 0 0 1 7 0 0 0 0 0 0 34 34 0 0 0 0 1 7

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,11,32,0,0,0,0,9,30,0,0],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;`

C4223D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{23}D_{10}`
`% in TeX`

`G:=Group("C4^2:23D10");`
`// GroupNames label`

`G:=SmallGroup(320,1376);`
`// by ID`

`G=gap.SmallGroup(320,1376);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,192,12550]);`
`// Polycyclic`

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

׿
×
𝔽