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## G = C10.1212+ 1+4order 320 = 26·5

### 30th 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 — C10.1212+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D10⋊C4 — C10.1212+ 1+4
 Lower central C5 — C2×C10 — C10.1212+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.1212+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a5b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1478 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×2], C22 [×27], C5, C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D5 [×6], C10 [×3], C10 [×3], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×3], C20 [×5], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×7], D20 [×7], C2×Dic5 [×4], C2×Dic5, C5⋊D4 [×4], C5⋊D4 [×5], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×4], C22×D5 [×10], C22×C10 [×2], D45D4, C4×Dic5, C10.D4 [×2], D10⋊C4 [×8], C23.D5, C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×4], C4○D20 [×4], D4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×4], C22×C20, D4×C10, C23×D5 [×2], D5×C22⋊C4, Dic54D4, C22⋊D20 [×2], D10⋊D4, Dic5.5D4, D208C4, D10.13D4, C4⋊D20, D10⋊Q8, C2×D10⋊C4, Dic5⋊D4, C5×C22.D4, C2×C4○D20, C2×D4×D5, C10.1212+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D45D4, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4, D48D10, C10.1212+ 1+4

Smallest permutation representation of C10.1212+ 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 50 28 38)(2 41 29 39)(3 42 30 40)(4 43 21 31)(5 44 22 32)(6 45 23 33)(7 46 24 34)(8 47 25 35)(9 48 26 36)(10 49 27 37)(11 58 73 70)(12 59 74 61)(13 60 75 62)(14 51 76 63)(15 52 77 64)(16 53 78 65)(17 54 79 66)(18 55 80 67)(19 56 71 68)(20 57 72 69)
(1 50 6 45)(2 49 7 44)(3 48 8 43)(4 47 9 42)(5 46 10 41)(11 65 16 70)(12 64 17 69)(13 63 18 68)(14 62 19 67)(15 61 20 66)(21 35 26 40)(22 34 27 39)(23 33 28 38)(24 32 29 37)(25 31 30 36)(51 80 56 75)(52 79 57 74)(53 78 58 73)(54 77 59 72)(55 76 60 71)
(1 53 28 65)(2 52 29 64)(3 51 30 63)(4 60 21 62)(5 59 22 61)(6 58 23 70)(7 57 24 69)(8 56 25 68)(9 55 26 67)(10 54 27 66)(11 45 73 33)(12 44 74 32)(13 43 75 31)(14 42 76 40)(15 41 77 39)(16 50 78 38)(17 49 79 37)(18 48 80 36)(19 47 71 35)(20 46 72 34)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 61)(8 62)(9 63)(10 64)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4×D5 D5×C4○D4 D4⋊8D10 kernel C10.1212+ 1+4 D5×C22⋊C4 Dic5⋊4D4 C22⋊D20 D10⋊D4 Dic5.5D4 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊Q8 C2×D10⋊C4 Dic5⋊D4 C5×C22.D4 C2×C4○D20 C2×D4×D5 C5⋊D4 C22.D4 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 4 2 4 6 4 2 2 1 4 4 4

Matrix representation of C10.1212+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 23 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 40 18 0 0 0 0 9 1 0 0 0 0 0 0 40 0 0 0 0 0 7 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 7 1 0 0 0 0 0 0 40 5 0 0 0 0 16 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 5 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,23,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,5,1] >;`

C10.1212+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{121}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.121ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1326);`
`// by ID`

`G=gap.SmallGroup(320,1326);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,346,297,12550]);`
`// Polycyclic`

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

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