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## G = C24.33D10order 320 = 26·5

### 33rd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.33D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C24.33D10
 Lower central C5 — C2×C10 — C24.33D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.33D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1286 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×4], C22 [×25], C5, C2×C4 [×3], C2×C4 [×16], D4 [×18], Q8 [×2], C23 [×4], C23 [×12], D5 [×3], C10 [×3], C10 [×6], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×2], Dic5 [×5], C20 [×3], D10 [×2], D10 [×9], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×3], C2×Dic5 [×6], C2×Dic5 [×7], C5⋊D4 [×4], C5⋊D4 [×9], C2×C20 [×3], C5×D4 [×5], C22×D5 [×2], C22×D5 [×5], C22×C10 [×4], C22×C10 [×5], D45D4, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×5], C5×C22⋊C4 [×3], C2×Dic10, C2×C4×D5 [×2], D42D5 [×4], C22×Dic5 [×4], C2×C5⋊D4 [×6], C2×C5⋊D4 [×4], D4×C10 [×3], C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic54D4, D10.12D4, Dic5.5D4, C22.D20, D4×Dic5, C23⋊D10, C202D4, Dic5⋊D4 [×2], C2×C23.D5, C5×C22≀C2, C2×D42D5, C22×C5⋊D4, C24.33D10
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, C22×D5 [×7], D45D4, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D46D10, C24.33D10

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

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D ··· 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 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 20 ··· 20 size 1 1 1 1 2 2 2 2 4 4 10 10 20 4 4 4 10 ··· 10 20 20 20 2 2 2 ··· 2 4 ··· 4 8 8 8 ··· 8

53 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 4 type + + + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 2+ 1+4 D4×D5 D4⋊2D5 D4⋊6D10 kernel C24.33D10 Dic5.14D4 D5×C22⋊C4 Dic5⋊4D4 D10.12D4 Dic5.5D4 C22.D20 D4×Dic5 C23⋊D10 C20⋊2D4 Dic5⋊D4 C2×C23.D5 C5×C22≀C2 C2×D4⋊2D5 C22×C5⋊D4 C5⋊D4 C22≀C2 C2×C10 C22⋊C4 C2×D4 C24 C10 C22 C22 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 4 2 4 6 6 2 1 4 4 4

Matrix representation of C24.33D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 2 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 34 34 0 0 0 0 7 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 32 0 0 0 0 0 32 9
,
 34 34 0 0 0 0 1 7 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,32,0,0,0,0,0,9],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;`

C24.33D10 in GAP, Magma, Sage, TeX

`C_2^4._{33}D_{10}`
`% in TeX`

`G:=Group("C2^4.33D10");`
`// GroupNames label`

`G:=SmallGroup(320,1263);`
`// by ID`

`G=gap.SmallGroup(320,1263);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,12550]);`
`// 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=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,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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