Copied to
clipboard

## G = C10.482+ 1+4order 320 = 26·5

### 48th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.482+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C23⋊D10 — C10.482+ 1+4
 Lower central C5 — C2×C10 — C10.482+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.482+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >

Subgroups: 1126 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C422C2, C41D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.54C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23.D10, C22⋊D20, D10⋊D4, C22.D20, D10.13D4, C4⋊C4⋊D5, C23.23D10, C207D4, C23⋊D10, C202D4, Dic5⋊D4, C20⋊D4, C5×C4⋊D4, C10.482+ 1+4
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D46D10, D48D10, C10.482+ 1+4

Smallest permutation representation of C10.482+ 1+4
On 80 points
Generators in S80
```(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 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)```

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

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

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

47 conjugacy classes

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

47 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 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D4⋊8D10 kernel C10.482+ 1+4 C23.D10 C22⋊D20 D10⋊D4 C22.D20 D10.13D4 C4⋊C4⋊D5 C23.23D10 C20⋊7D4 C23⋊D10 C20⋊2D4 Dic5⋊D4 C20⋊D4 C5×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 2 1 1 2 4 2 2 6 3 8 4

Matrix representation of C10.482+ 1+4 in GL8(𝔽41)

 1 34 0 0 0 0 0 0 7 34 0 0 0 0 0 0 0 0 1 34 0 0 0 0 0 0 7 34 0 0 0 0 0 0 0 0 35 35 0 0 0 0 0 0 6 40 0 0 0 0 0 0 0 0 0 35 0 0 0 0 0 0 7 34
,
 24 40 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 24 40 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 1 0 38 21 0 0 0 0 0 1 40 21 0 0 0 0 1 40 40 0 0 0 0 0 2 35 0 40
,
 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 40 40 0 0 0 0 0 2 35 0 40
,
 40 7 40 7 0 0 0 0 0 1 0 1 0 0 0 0 2 27 1 34 0 0 0 0 0 39 0 40 0 0 0 0 0 0 0 0 23 6 0 0 0 0 0 0 21 18 0 0 0 0 0 0 0 0 38 23 0 0 0 0 0 0 37 3
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 39 0 40 0 0 0 0 0 0 39 0 40 0 0 0 0 0 0 0 0 23 6 0 0 0 0 0 0 35 18 0 0 0 0 0 0 0 0 17 6 0 0 0 0 0 0 34 24

`G:=sub<GL(8,GF(41))| [1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34],[24,1,0,0,0,0,0,0,40,17,0,0,0,0,0,0,0,0,24,1,0,0,0,0,0,0,40,17,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,1,40,35,0,0,0,0,38,40,40,0,0,0,0,0,21,21,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,1,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,2,0,0,0,0,0,7,1,27,39,0,0,0,0,40,0,1,0,0,0,0,0,7,1,34,40,0,0,0,0,0,0,0,0,23,21,0,0,0,0,0,0,6,18,0,0,0,0,0,0,0,0,38,37,0,0,0,0,0,0,23,3],[1,0,39,0,0,0,0,0,0,1,0,39,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,23,35,0,0,0,0,0,0,6,18,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24] >;`

C10.482+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{48}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.48ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1292);`
`// by ID`

`G=gap.SmallGroup(320,1292);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=a^5*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;`
`// generators/relations`

׿
×
𝔽