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## G = C10.382+ 1+4order 320 = 26·5

### 38th 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 — C10.382+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C10.382+ 1+4
 Lower central C5 — C2×C10 — C10.382+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.382+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=a5c, ede=b2d >

Subgroups: 1534 in 334 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×12], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], D5 [×5], C10 [×3], C10 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×19], C2×C10, C2×C10 [×9], C42⋊C2, C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], C4×D5 [×2], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5 [×2], C22×D5 [×2], C22×D5 [×8], C22×C10, C22×C10 [×2], C22.29C24, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×2], C4○D20 [×4], D4×D5 [×4], C2×C5⋊D4 [×2], C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], C23×D5 [×2], C22⋊D20 [×2], Dic5.5D4 [×2], C4⋊C47D5, C4⋊D20, C207D4, C23⋊D10 [×2], C202D4, C20⋊D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D4×D5, C10.382+ 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], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D48D10, C10.382+ 1+4

Smallest permutation representation of C10.382+ 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 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 38 13 28)(2 37 14 27)(3 36 15 26)(4 35 16 25)(5 34 17 24)(6 33 18 23)(7 32 19 22)(8 31 20 21)(9 40 11 30)(10 39 12 29)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(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), (1,43,13,53)(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)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,13,28)(2,37,14,27)(3,36,15,26)(4,35,16,25)(5,34,17,24)(6,33,18,23)(7,32,19,22)(8,31,20,21)(9,40,11,30)(10,39,12,29)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(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), (1,43,13,53)(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)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,13,28)(2,37,14,27)(3,36,15,26)(4,35,16,25)(5,34,17,24)(6,33,18,23)(7,32,19,22)(8,31,20,21)(9,40,11,30)(10,39,12,29)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(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)], [(1,43,13,53),(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),(21,66,31,76),(22,67,32,77),(23,68,33,78),(24,69,34,79),(25,70,35,80),(26,61,36,71),(27,62,37,72),(28,63,38,73),(29,64,39,74),(30,65,40,75)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,38,13,28),(2,37,14,27),(3,36,15,26),(4,35,16,25),(5,34,17,24),(6,33,18,23),(7,32,19,22),(8,31,20,21),(9,40,11,30),(10,39,12,29),(41,75,51,65),(42,74,52,64),(43,73,53,63),(44,72,54,62),(45,71,55,61),(46,80,56,70),(47,79,57,69),(48,78,58,68),(49,77,59,67),(50,76,60,66)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)])`

50 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 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 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 4 4 4 10 10 20 20 20 2 2 4 4 4 10 10 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 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 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 C10.382+ 1+4 C22⋊D20 Dic5.5D4 C4⋊C4⋊7D5 C4⋊D20 C20⋊7D4 C23⋊D10 C20⋊2D4 C20⋊D4 C5×C4⋊D4 C2×C4○D20 C2×D4×D5 C4×D5 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C2 C2 # reps 1 2 2 1 1 1 2 1 2 1 1 1 4 2 4 2 2 6 2 4 4 4

Matrix representation of C10.382+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 18 35 2 0 0 0 6 23 0 2 0 0 40 0 23 6 0 0 0 40 35 18
,
 1 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 23 6 40 0 0 0 35 18 0 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 18 35 0 0 0 0 20 23 0 0 0 0 1 0 18 35 0 0 6 40 20 23
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 18 35 0 0 0 0 6 23 0 0 0 0 0 0 18 35 0 0 0 0 6 23

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,6,40,0,0,0,35,23,0,40,0,0,2,0,23,35,0,0,0,2,6,18],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,23,35,0,0,0,1,6,18,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,20,1,6,0,0,35,23,0,40,0,0,0,0,18,20,0,0,0,0,35,23],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,0,0,0,0,18,6,0,0,0,0,35,23] >;`

C10.382+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{38}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.38ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1279);`
`// by ID`

`G=gap.SmallGroup(320,1279);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,12550]);`
`// Polycyclic`

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

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