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## G = C10.462+ 1+4order 320 = 26·5

### 46th 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 — C10.462+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C23⋊D10 — C10.462+ 1+4
 Lower central C5 — C2×C10 — C10.462+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.462+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1006 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×9], Q8, C23, C23 [×2], C23 [×6], D5 [×2], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C24, Dic5 [×6], C20 [×4], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2 [×2], Dic10, C2×Dic5 [×6], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×D5 [×2], C22×D5 [×4], C22×C10, C22×C10 [×2], C22.32C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], C23×D5, Dic5.5D4 [×2], C22.D20 [×2], C4⋊C4⋊D5 [×2], C20.48D4, C2×D10⋊C4, D4×Dic5 [×2], C23⋊D10 [×2], Dic5⋊D4 [×2], C5×C4⋊D4, C10.462+ 1+4
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, D42D5 [×2], C23×D5, C2×D42D5, D46D10, D48D10, C10.462+ 1+4

Smallest permutation representation of C10.462+ 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 13 53)(2 44 14 54)(3 45 15 55)(4 46 16 56)(5 47 17 57)(6 48 18 58)(7 49 19 59)(8 50 20 60)(9 41 11 51)(10 42 12 52)(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)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 75 56 70)(42 74 57 69)(43 73 58 68)(44 72 59 67)(45 71 60 66)(46 80 51 65)(47 79 52 64)(48 78 53 63)(49 77 54 62)(50 76 55 61)
(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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(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), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61), (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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(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), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61), (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)], [(1,43,13,53),(2,44,14,54),(3,45,15,55),(4,46,16,56),(5,47,17,57),(6,48,18,58),(7,49,19,59),(8,50,20,60),(9,41,11,51),(10,42,12,52),(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)], [(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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),(41,75,56,70),(42,74,57,69),(43,73,58,68),(44,72,59,67),(45,71,60,66),(46,80,51,65),(47,79,52,64),(48,78,53,63),(49,77,54,62),(50,76,55,61)], [(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 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 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 2 2 4 4 20 20 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

50 irreducible representations

 dim 1 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 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 D4⋊8D10 kernel C10.462+ 1+4 Dic5.5D4 C22.D20 C4⋊C4⋊D5 C20.48D4 C2×D10⋊C4 D4×Dic5 C23⋊D10 Dic5⋊D4 C5×C4⋊D4 C4⋊D4 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 2 2 2 1 1 2 2 2 1 2 4 4 2 2 6 2 4 4 4

Matrix representation of C10.462+ 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 1 34 0 0 0 0 7 34
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 27 30 0 0 0 1 27 27 0 0 28 19 40 0 0 0 13 28 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 28 19 40 0 0 0 13 28 0 40
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 11 9 0 0 0 0 14 30 0 0 0 0 0 0 14 14 0 0 0 0 30 27
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 0 0 14 2 11 32 0 0 14 14 9 30

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,1,7,0,0,0,0,34,34],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,28,13,0,0,0,1,19,28,0,0,27,27,40,0,0,0,30,27,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,28,13,0,0,0,1,19,28,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,11,14,0,0,0,0,9,30,0,0,0,0,0,0,14,30,0,0,0,0,14,27],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,11,32,14,14,0,0,9,30,2,14,0,0,0,0,11,9,0,0,0,0,32,30] >;`

C10.462+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{46}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.46ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1289);`
`// by ID`

`G=gap.SmallGroup(320,1289);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,675,297,136,12550]);`
`// Polycyclic`

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

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