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## G = C24.31D10order 320 = 26·5

### 31st non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.31D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C24.31D10
 Lower central C5 — C2×C10 — C24.31D10
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C24.31D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 806 in 248 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4 [×4], C2×C4 [×14], D4 [×5], Q8, C23 [×3], C23 [×6], D5, C10 [×3], C10 [×5], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C24, Dic5 [×7], C20 [×4], D10 [×3], C2×C10, C2×C10 [×4], C2×C10 [×11], C2×C22⋊C4, C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×5], C2×C20 [×4], C2×C20 [×3], C22×D5, C22×C10 [×3], C22×C10 [×5], C22.45C24, C4×Dic5 [×3], C10.D4 [×5], C4⋊Dic5 [×3], D10⋊C4 [×3], C23.D5 [×7], C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×3], C22×C20 [×2], C23×C10, C23.11D10, Dic5.14D4, C23.D10 [×2], Dic54D4, D10.12D4, Dic5.5D4, C22.D20, C20.48D4, C23.21D10, C4×C5⋊D4, C23.23D10, C2×C23.D5, C242D5, C10×C22⋊C4, C24.31D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.45C24, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D46D10, C24.31D10

Smallest permutation representation of C24.31D10
On 80 points
Generators in S80
```(2 50)(4 52)(6 54)(8 56)(10 58)(12 60)(14 42)(16 44)(18 46)(20 48)(21 31)(22 76)(23 33)(24 78)(25 35)(26 80)(27 37)(28 62)(29 39)(30 64)(32 66)(34 68)(36 70)(38 72)(40 74)(61 71)(63 73)(65 75)(67 77)(69 79)
(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(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 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 28 11 38)(2 37 12 27)(3 26 13 36)(4 35 14 25)(5 24 15 34)(6 33 16 23)(7 22 17 32)(8 31 18 21)(9 40 19 30)(10 29 20 39)(41 80 51 70)(42 69 52 79)(43 78 53 68)(44 67 54 77)(45 76 55 66)(46 65 56 75)(47 74 57 64)(48 63 58 73)(49 72 59 62)(50 61 60 71)```

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4O 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P 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 20 ··· 20 size 1 1 1 1 2 2 2 2 4 20 2 2 2 2 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 2+ 1+4 D4⋊2D5 D4⋊6D10 kernel C24.31D10 C23.11D10 Dic5.14D4 C23.D10 Dic5⋊4D4 D10.12D4 Dic5.5D4 C22.D20 C20.48D4 C23.21D10 C4×C5⋊D4 C23.23D10 C2×C23.D5 C24⋊2D5 C10×C22⋊C4 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C22 C10 C22 C2 # reps 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 8 8 4 2 16 1 4 4

Matrix representation of C24.31D10 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 1 0 0 0 24 40
,
 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 21 0 0 0 0 39 0 0 0 0 1 17 0 0 24 40
,
 0 39 0 0 21 0 0 0 0 0 32 0 0 0 0 32
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,24,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[21,0,0,0,0,39,0,0,0,0,1,24,0,0,17,40],[0,21,0,0,39,0,0,0,0,0,32,0,0,0,0,32] >;`

C24.31D10 in GAP, Magma, Sage, TeX

`C_2^4._{31}D_{10}`
`% in TeX`

`G:=Group("C2^4.31D10");`
`// GroupNames label`

`G:=SmallGroup(320,1167);`
`// by ID`

`G=gap.SmallGroup(320,1167);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;`
`// generators/relations`

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