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## G = C10.512+ 1+4order 320 = 26·5

### 51st 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 — C10.512+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C10.512+ 1+4
 Lower central C5 — C2×C10 — C10.512+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.512+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 958 in 242 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], Q8 [×4], C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×Q8, C2×Q8 [×3], C24, Dic5 [×6], C20 [×6], D10 [×4], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×3], C22⋊Q8, C22⋊Q8 [×11], Dic10 [×3], C4×D5 [×4], C2×Dic5 [×6], C2×Dic5, C2×C20 [×2], C2×C20 [×4], C2×C20, C5×Q8, C22×D5 [×6], C22×D5 [×2], C22×C10, C232Q8, C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×4], C22×Dic5, C22×C20, Q8×C10, C23×D5, Dic5.14D4 [×2], D5×C22⋊C4 [×2], D10⋊Q8 [×4], D102Q8 [×2], C20.48D4, C2×D10⋊C4, D103Q8 [×2], C5×C22⋊Q8, C10.512+ 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4 [×2], C22×D5 [×7], C232Q8, Q8×D5 [×2], C23×D5, D46D10, C2×Q8×D5, D48D10, C10.512+ 1+4

Smallest permutation representation of C10.512+ 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 43 18 58)(2 44 19 59)(3 45 20 60)(4 46 11 51)(5 47 12 52)(6 48 13 53)(7 49 14 54)(8 50 15 55)(9 41 16 56)(10 42 17 57)(21 61 36 76)(22 62 37 77)(23 63 38 78)(24 64 39 79)(25 65 40 80)(26 66 31 71)(27 67 32 72)(28 68 33 73)(29 69 34 74)(30 70 35 75)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 33 18 28)(2 32 19 27)(3 31 20 26)(4 40 11 25)(5 39 12 24)(6 38 13 23)(7 37 14 22)(8 36 15 21)(9 35 16 30)(10 34 17 29)(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 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(21,61,36,76)(22,62,37,77)(23,63,38,78)(24,64,39,79)(25,65,40,80)(26,66,31,71)(27,67,32,72)(28,68,33,73)(29,69,34,74)(30,70,35,75), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,33,18,28)(2,32,19,27)(3,31,20,26)(4,40,11,25)(5,39,12,24)(6,38,13,23)(7,37,14,22)(8,36,15,21)(9,35,16,30)(10,34,17,29)(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), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)>;`

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 D10 D10 D10 D10 2+ 1+4 Q8×D5 D4⋊6D10 D4⋊8D10 kernel C10.512+ 1+4 Dic5.14D4 D5×C22⋊C4 D10⋊Q8 D10⋊2Q8 C20.48D4 C2×D10⋊C4 D10⋊3Q8 C5×C22⋊Q8 C22×D5 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C22 C2 C2 # reps 1 2 2 4 2 1 1 2 1 4 2 4 6 2 2 2 4 4 4

Matrix representation of C10.512+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 34 34 0 0 0 0 7 1 0 0 0 0 0 0 34 34 0 0 0 0 7 1
,
 12 5 0 0 0 0 12 29 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 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 4 0 0 0 0 5 15 0 0 0 0 0 0 30 32 0 0 0 0 27 11 0 0 0 0 0 0 11 9 0 0 0 0 14 30
,
 26 4 0 0 0 0 5 15 0 0 0 0 0 0 30 32 0 0 0 0 9 11 0 0 0 0 0 0 11 9 0 0 0 0 32 30

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[12,12,0,0,0,0,5,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,5,0,0,0,0,4,15,0,0,0,0,0,0,30,27,0,0,0,0,32,11,0,0,0,0,0,0,11,14,0,0,0,0,9,30],[26,5,0,0,0,0,4,15,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,11,32,0,0,0,0,9,30] >;`

C10.512+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{51}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.51ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1306);`
`// by ID`

`G=gap.SmallGroup(320,1306);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,297,136,12550]);`
`// Polycyclic`

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

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