Copied to
clipboard

## G = C10.1462+ 1+4order 320 = 26·5

### 55th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.1462+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C10.1462+ 1+4
 Lower central C5 — C2×C10 — C10.1462+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C10.1462+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a5b-1, dbd-1=a5b, be=eb, cd=dc, ece=a5c, ede=a5b2d >

Subgroups: 1454 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C22×D4, C2×C4○D4, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C22×C10, C22.29C24, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C2×D20, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C23×D5, C23.21D10, C207D4, C23⋊D10, C20⋊D4, C20.23D4, C22×D20, C10×C4○D4, C10.1462+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C5⋊D4, C22×D5, C22.29C24, C2×C5⋊D4, C23×D5, D48D10, C22×C5⋊D4, C10.1462+ 1+4

Smallest permutation representation of C10.1462+ 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 11 30 75)(2 12 21 76)(3 13 22 77)(4 14 23 78)(5 15 24 79)(6 16 25 80)(7 17 26 71)(8 18 27 72)(9 19 28 73)(10 20 29 74)(31 66 41 56)(32 67 42 57)(33 68 43 58)(34 69 44 59)(35 70 45 60)(36 61 46 51)(37 62 47 52)(38 63 48 53)(39 64 49 54)(40 65 50 55)
(1 75 6 80)(2 74 7 79)(3 73 8 78)(4 72 9 77)(5 71 10 76)(11 25 16 30)(12 24 17 29)(13 23 18 28)(14 22 19 27)(15 21 20 26)(31 59 36 54)(32 58 37 53)(33 57 38 52)(34 56 39 51)(35 55 40 60)(41 69 46 64)(42 68 47 63)(43 67 48 62)(44 66 49 61)(45 65 50 70)
(1 50 25 35)(2 49 26 34)(3 48 27 33)(4 47 28 32)(5 46 29 31)(6 45 30 40)(7 44 21 39)(8 43 22 38)(9 42 23 37)(10 41 24 36)(11 60 80 65)(12 59 71 64)(13 58 72 63)(14 57 73 62)(15 56 74 61)(16 55 75 70)(17 54 76 69)(18 53 77 68)(19 52 78 67)(20 51 79 66)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 31)(8 32)(9 33)(10 34)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(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), (1,11,30,75)(2,12,21,76)(3,13,22,77)(4,14,23,78)(5,15,24,79)(6,16,25,80)(7,17,26,71)(8,18,27,72)(9,19,28,73)(10,20,29,74)(31,66,41,56)(32,67,42,57)(33,68,43,58)(34,69,44,59)(35,70,45,60)(36,61,46,51)(37,62,47,52)(38,63,48,53)(39,64,49,54)(40,65,50,55), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26)(31,59,36,54)(32,58,37,53)(33,57,38,52)(34,56,39,51)(35,55,40,60)(41,69,46,64)(42,68,47,63)(43,67,48,62)(44,66,49,61)(45,65,50,70), (1,50,25,35)(2,49,26,34)(3,48,27,33)(4,47,28,32)(5,46,29,31)(6,45,30,40)(7,44,21,39)(8,43,22,38)(9,42,23,37)(10,41,24,36)(11,60,80,65)(12,59,71,64)(13,58,72,63)(14,57,73,62)(15,56,74,61)(16,55,75,70)(17,54,76,69)(18,53,77,68)(19,52,78,67)(20,51,79,66), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,31)(8,32)(9,33)(10,34)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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), (1,11,30,75)(2,12,21,76)(3,13,22,77)(4,14,23,78)(5,15,24,79)(6,16,25,80)(7,17,26,71)(8,18,27,72)(9,19,28,73)(10,20,29,74)(31,66,41,56)(32,67,42,57)(33,68,43,58)(34,69,44,59)(35,70,45,60)(36,61,46,51)(37,62,47,52)(38,63,48,53)(39,64,49,54)(40,65,50,55), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26)(31,59,36,54)(32,58,37,53)(33,57,38,52)(34,56,39,51)(35,55,40,60)(41,69,46,64)(42,68,47,63)(43,67,48,62)(44,66,49,61)(45,65,50,70), (1,50,25,35)(2,49,26,34)(3,48,27,33)(4,47,28,32)(5,46,29,31)(6,45,30,40)(7,44,21,39)(8,43,22,38)(9,42,23,37)(10,41,24,36)(11,60,80,65)(12,59,71,64)(13,58,72,63)(14,57,73,62)(15,56,74,61)(16,55,75,70)(17,54,76,69)(18,53,77,68)(19,52,78,67)(20,51,79,66), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,31)(8,32)(9,33)(10,34)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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)], [(1,11,30,75),(2,12,21,76),(3,13,22,77),(4,14,23,78),(5,15,24,79),(6,16,25,80),(7,17,26,71),(8,18,27,72),(9,19,28,73),(10,20,29,74),(31,66,41,56),(32,67,42,57),(33,68,43,58),(34,69,44,59),(35,70,45,60),(36,61,46,51),(37,62,47,52),(38,63,48,53),(39,64,49,54),(40,65,50,55)], [(1,75,6,80),(2,74,7,79),(3,73,8,78),(4,72,9,77),(5,71,10,76),(11,25,16,30),(12,24,17,29),(13,23,18,28),(14,22,19,27),(15,21,20,26),(31,59,36,54),(32,58,37,53),(33,57,38,52),(34,56,39,51),(35,55,40,60),(41,69,46,64),(42,68,47,63),(43,67,48,62),(44,66,49,61),(45,65,50,70)], [(1,50,25,35),(2,49,26,34),(3,48,27,33),(4,47,28,32),(5,46,29,31),(6,45,30,40),(7,44,21,39),(8,43,22,38),(9,42,23,37),(10,41,24,36),(11,60,80,65),(12,59,71,64),(13,58,72,63),(14,57,73,62),(15,56,74,61),(16,55,75,70),(17,54,76,69),(18,53,77,68),(19,52,78,67),(20,51,79,66)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,31),(8,32),(9,33),(10,34),(11,70),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,46),(22,47),(23,48),(24,49),(25,50),(26,41),(27,42),(28,43),(29,44),(30,45),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)]])`

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 C5⋊D4 2+ 1+4 D4⋊8D10 kernel C10.1462+ 1+4 C23.21D10 C20⋊7D4 C23⋊D10 C20⋊D4 C20.23D4 C22×D20 C10×C4○D4 C2×C20 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C2×C4 C10 C2 # reps 1 1 4 4 2 2 1 1 4 2 6 6 2 16 2 8

Matrix representation of C10.1462+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 6 0 0 0 0 34 7 0 0 0 0 0 0 0 6 0 0 0 0 34 7
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 28 0 0 0 0 22 30 0 0 11 28 0 0 0 0 22 30 0 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 14 2 0 0 0 0 5 27 0 0 27 39 0 0 0 0 36 14 0 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 7 40 0 0 0 0 7 34 0 0 7 40 0 0 0 0 7 34 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 40 0 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,0,34,0,0,0,0,6,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,22,0,0,0,0,28,30,0,0,11,22,0,0,0,0,28,30,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,36,0,0,0,0,39,14,0,0,14,5,0,0,0,0,2,27,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,7,7,0,0,0,0,40,34,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0] >;`

C10.1462+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{146}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.146ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1502);`
`// by ID`

`G=gap.SmallGroup(320,1502);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,570,12550]);`
`// Polycyclic`

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

׿
×
𝔽