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G = C10.1452+ 1+4order 320 = 26·5

54th 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 — C10.1452+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C10.1452+ 1+4
 Lower central C5 — C2×C10 — C10.1452+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C10.1452+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 1334 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×13], D5 [×4], C10 [×3], C10 [×5], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×2], D10 [×16], C2×C10, C2×C10 [×4], C2×C10 [×7], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×6], C5×D4 [×4], C5×D4 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×D5 [×10], C22×C10, C22×C10 [×2], D45D4, C4×Dic5, C10.D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4, D10⋊C4 [×8], C23.D5, C23.D5 [×2], C2×C4×D5, C2×D20, D4×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C23×D5 [×2], C2×D10⋊C4 [×2], C4×C5⋊D4, C23.23D10 [×2], C207D4, D4×Dic5, C23⋊D10 [×2], Dic5⋊D4 [×2], D103Q8, C20.23D4, C2×D4×D5, C10×C4○D4, C10.1452+ 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, C5⋊D4 [×4], C22×D5 [×7], D45D4, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4, D48D10, C22×C5⋊D4, C10.1452+ 1+4

Smallest permutation representation of C10.1452+ 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 11 23 78)(2 12 24 79)(3 13 25 80)(4 14 26 71)(5 15 27 72)(6 16 28 73)(7 17 29 74)(8 18 30 75)(9 19 21 76)(10 20 22 77)(31 68 48 51)(32 69 49 52)(33 70 50 53)(34 61 41 54)(35 62 42 55)(36 63 43 56)(37 64 44 57)(38 65 45 58)(39 66 46 59)(40 67 47 60)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 16)(12 17)(13 18)(14 19)(15 20)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 38 23 45)(2 39 24 46)(3 40 25 47)(4 31 26 48)(5 32 27 49)(6 33 28 50)(7 34 29 41)(8 35 30 42)(9 36 21 43)(10 37 22 44)(11 65 78 58)(12 66 79 59)(13 67 80 60)(14 68 71 51)(15 69 72 52)(16 70 73 53)(17 61 74 54)(18 62 75 55)(19 63 76 56)(20 64 77 57)
(1 38 6 33)(2 37 7 32)(3 36 8 31)(4 35 9 40)(5 34 10 39)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 47 26 42)(22 46 27 41)(23 45 28 50)(24 44 29 49)(25 43 30 48)(51 75 56 80)(52 74 57 79)(53 73 58 78)(54 72 59 77)(55 71 60 76)```

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

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

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

65 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 ··· 10R 20A ··· 20H 20I ··· 20T 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 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 4 10 10 20 20 2 2 2 2 4 4 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 2+ 1+4 D5×C4○D4 D4⋊8D10 kernel C10.1452+ 1+4 C2×D10⋊C4 C4×C5⋊D4 C23.23D10 C20⋊7D4 D4×Dic5 C23⋊D10 Dic5⋊D4 D10⋊3Q8 C20.23D4 C2×D4×D5 C10×C4○D4 C5×D4 C2×C4○D4 D10 C22×C4 C2×D4 C2×Q8 D4 C10 C2 C2 # reps 1 2 1 2 1 1 2 2 1 1 1 1 4 2 4 6 6 2 16 1 4 4

Matrix representation of C10.1452+ 1+4 in GL4(𝔽41) generated by

 7 6 0 0 34 0 0 0 0 0 40 0 0 0 0 40
,
 17 35 0 0 7 24 0 0 0 0 32 23 0 0 0 9
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 40 40
,
 24 6 0 0 34 17 0 0 0 0 32 0 0 0 0 32
,
 3 21 0 0 21 38 0 0 0 0 32 0 0 0 9 9
`G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,40,0,0,0,0,40],[17,7,0,0,35,24,0,0,0,0,32,0,0,0,23,9],[40,0,0,0,0,40,0,0,0,0,1,40,0,0,0,40],[24,34,0,0,6,17,0,0,0,0,32,0,0,0,0,32],[3,21,0,0,21,38,0,0,0,0,32,9,0,0,0,9] >;`

C10.1452+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{145}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.145ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1501);`
`// by ID`

`G=gap.SmallGroup(320,1501);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,136,12550]);`
`// 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,a*d=d*a,e*a*e^-1=a^-1,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=a^5*b^2*d>;`
`// generators/relations`

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