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## G = C10.372+ 1+4order 320 = 26·5

### 37th 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 — C10.372+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C10.372+ 1+4
 Lower central C5 — C2×C10 — C10.372+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.372+ 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, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1630 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×34], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×20], C23, C23 [×2], C23 [×18], D5 [×6], C10 [×3], C10 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×17], C24 [×3], Dic5 [×4], C20 [×4], D10 [×4], D10 [×22], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×4], C22×D4 [×2], C4×D5 [×4], D20 [×5], C2×Dic5 [×4], C2×Dic5, C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×D5 [×8], C22×D5 [×10], C22×C10, C22×C10 [×2], C233D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], D4×D5 [×8], C22×Dic5, C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], C23×D5, C23×D5 [×2], C22⋊D20 [×2], D10.12D4 [×2], D10.13D4 [×2], C2×D10⋊C4, C207D4, C23⋊D10 [×2], Dic5⋊D4 [×2], C5×C4⋊D4, C2×D4×D5 [×2], C10.372+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C22×D5 [×7], C233D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D48D10, C10.372+ 1+4

Smallest permutation representation of C10.372+ 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 78 13 68)(2 79 14 69)(3 80 15 70)(4 71 16 61)(5 72 17 62)(6 73 18 63)(7 74 19 64)(8 75 20 65)(9 76 11 66)(10 77 12 67)(21 56 31 46)(22 57 32 47)(23 58 33 48)(24 59 34 49)(25 60 35 50)(26 51 36 41)(27 52 37 42)(28 53 38 43)(29 54 39 44)(30 55 40 45)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 41)(10 42)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 78 18 63)(2 77 19 62)(3 76 20 61)(4 75 11 70)(5 74 12 69)(6 73 13 68)(7 72 14 67)(8 71 15 66)(9 80 16 65)(10 79 17 64)(21 45 36 60)(22 44 37 59)(23 43 38 58)(24 42 39 57)(25 41 40 56)(26 50 31 55)(27 49 32 54)(28 48 33 53)(29 47 34 52)(30 46 35 51)
(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 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 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), (1,78,13,68)(2,79,14,69)(3,80,15,70)(4,71,16,61)(5,72,17,62)(6,73,18,63)(7,74,19,64)(8,75,20,65)(9,76,11,66)(10,77,12,67)(21,56,31,46)(22,57,32,47)(23,58,33,48)(24,59,34,49)(25,60,35,50)(26,51,36,41)(27,52,37,42)(28,53,38,43)(29,54,39,44)(30,55,40,45), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,41)(10,42)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,78,18,63)(2,77,19,62)(3,76,20,61)(4,75,11,70)(5,74,12,69)(6,73,13,68)(7,72,14,67)(8,71,15,66)(9,80,16,65)(10,79,17,64)(21,45,36,60)(22,44,37,59)(23,43,38,58)(24,42,39,57)(25,41,40,56)(26,50,31,55)(27,49,32,54)(28,48,33,53)(29,47,34,52)(30,46,35,51), (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)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)>;`

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 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 2 2 2 2 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 10 10 10 10 20 20 4 4 4 4 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 D4 D5 D10 D10 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 D4⋊8D10 kernel C10.372+ 1+4 C22⋊D20 D10.12D4 D10.13D4 C2×D10⋊C4 C20⋊7D4 C23⋊D10 Dic5⋊D4 C5×C4⋊D4 C2×D4×D5 C22×D5 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 2 2 2 1 1 2 2 1 2 4 2 4 2 2 6 2 4 4 4

Matrix representation of C10.372+ 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
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 0 0 0 0 30 32 0 0 0 0 9 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 30 32 0 0 0 0 9 11 0 0
,
 0 40 0 0 0 0 1 0 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
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

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

C10.372+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{37}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.37ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1277);`
`// by ID`

`G=gap.SmallGroup(320,1277);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,136,12550]);`
`// 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,d*b*d^-1=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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