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## G = D5.2- 1+4order 320 = 26·5

### The non-split extension by D5 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5.2- 1+4
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C4×F5 — Q8×F5 — D5.2- 1+4
 Lower central C5 — C10 — D5.2- 1+4
 Upper central C1 — C2 — C2×Q8

Generators and relations for D5.2- 1+4
G = < a,b,c,d,e,f | a5=b2=c4=1, d2=a-1b, e2=a-1bc2, f2=c2, bab=cac-1=a-1, dad-1=eae-1=a3, af=fa, cbc-1=a3b, dbd-1=ebe-1=a2b, bf=fb, dcd-1=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 794 in 266 conjugacy classes, 138 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, F5, D10, D10, C2×C10, C42⋊C2, C4×Q8, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C2×F5, C22×D5, C23.32C23, C4×F5, C4⋊F5, C22⋊F5, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, D10.C23, Q8×F5, C2×Q8×D5, D5.2- 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, 2- 1+4, C2×F5, C23.32C23, C22×F5, C23×F5, D5.2- 1+4

Smallest permutation representation of D5.2- 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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)
(1 59 9 54)(2 58 10 53)(3 57 6 52)(4 56 7 51)(5 60 8 55)(11 47 16 42)(12 46 17 41)(13 50 18 45)(14 49 19 44)(15 48 20 43)(21 77 26 72)(22 76 27 71)(23 80 28 75)(24 79 29 74)(25 78 30 73)(31 67 36 62)(32 66 37 61)(33 70 38 65)(34 69 39 64)(35 68 40 63)
(1 54)(2 51 5 52)(3 53 4 55)(6 58 7 60)(8 57 10 56)(9 59)(11 48 12 50)(13 47 15 46)(14 49)(16 43 17 45)(18 42 20 41)(19 44)(21 73 22 75)(23 72 25 71)(24 74)(26 78 27 80)(28 77 30 76)(29 79)(31 68 32 70)(33 67 35 66)(34 69)(36 63 37 65)(38 62 40 61)(39 64)
(1 24 9 29)(2 21 8 27)(3 23 7 30)(4 25 6 28)(5 22 10 26)(11 38 17 35)(12 40 16 33)(13 37 20 31)(14 39 19 34)(15 36 18 32)(41 68 47 65)(42 70 46 63)(43 67 50 61)(44 69 49 64)(45 66 48 62)(51 73 57 80)(52 75 56 78)(53 72 60 76)(54 74 59 79)(55 71 58 77)
(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79), (1,59,9,54)(2,58,10,53)(3,57,6,52)(4,56,7,51)(5,60,8,55)(11,47,16,42)(12,46,17,41)(13,50,18,45)(14,49,19,44)(15,48,20,43)(21,77,26,72)(22,76,27,71)(23,80,28,75)(24,79,29,74)(25,78,30,73)(31,67,36,62)(32,66,37,61)(33,70,38,65)(34,69,39,64)(35,68,40,63), (1,54)(2,51,5,52)(3,53,4,55)(6,58,7,60)(8,57,10,56)(9,59)(11,48,12,50)(13,47,15,46)(14,49)(16,43,17,45)(18,42,20,41)(19,44)(21,73,22,75)(23,72,25,71)(24,74)(26,78,27,80)(28,77,30,76)(29,79)(31,68,32,70)(33,67,35,66)(34,69)(36,63,37,65)(38,62,40,61)(39,64), (1,24,9,29)(2,21,8,27)(3,23,7,30)(4,25,6,28)(5,22,10,26)(11,38,17,35)(12,40,16,33)(13,37,20,31)(14,39,19,34)(15,36,18,32)(41,68,47,65)(42,70,46,63)(43,67,50,61)(44,69,49,64)(45,66,48,62)(51,73,57,80)(52,75,56,78)(53,72,60,76)(54,74,59,79)(55,71,58,77), (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)>;`

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G ··· 4AB 5 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 10 10 10 20 ··· 20 size 1 1 2 5 5 10 2 ··· 2 10 ··· 10 4 4 4 4 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 4 4 8 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 C4 F5 2- 1+4 C2×F5 C2×F5 D5.2- 1+4 kernel D5.2- 1+4 D10.C23 Q8×F5 C2×Q8×D5 C2×Dic10 Q8×D5 Q8×C10 C2×Q8 D5 C2×C4 Q8 C1 # reps 1 6 8 1 6 8 2 1 2 3 4 2

Matrix representation of D5.2- 1+4 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 1
,
 13 36 5 10 0 0 0 0 32 13 27 15 0 0 0 0 38 0 1 2 0 0 0 0 21 13 27 14 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 6 4 37 33 0 0 0 0 19 6 26 29 0 0 0 0 27 0 32 23 0 0 0 0 16 6 35 38 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40
,
 40 0 0 0 0 0 0 0 38 1 0 0 0 0 0 0 16 0 1 0 0 0 0 0 11 1 40 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40
,
 40 0 5 0 0 0 0 0 0 1 27 39 0 0 0 0 16 0 1 0 0 0 0 0 11 1 27 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1],[13,32,38,21,0,0,0,0,36,13,0,13,0,0,0,0,5,27,1,27,0,0,0,0,10,15,2,14,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,1,0],[6,19,27,16,0,0,0,0,4,6,0,6,0,0,0,0,37,26,32,35,0,0,0,0,33,29,23,38,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40],[40,38,16,11,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40],[40,0,16,11,0,0,0,0,0,1,0,1,0,0,0,0,5,27,1,27,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

D5.2- 1+4 in GAP, Magma, Sage, TeX

`D_5.2_-^{1+4}`
`% in TeX`

`G:=Group("D5.ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1600);`
`// by ID`

`G=gap.SmallGroup(320,1600);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,6278,818]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=1,d^2=a^-1*b,e^2=a^-1*b*c^2,f^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=e*a*e^-1=a^3,a*f=f*a,c*b*c^-1=a^3*b,d*b*d^-1=e*b*e^-1=a^2*b,b*f=f*b,d*c*d^-1=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c^2*e>;`
`// generators/relations`

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