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## G = C24.24D10order 320 = 26·5

### 24th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C24.24D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C24.24D10
 Lower central C5 — C10 — C24.24D10
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C24.24D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1166 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C22, C22 [×6], C22 [×22], C5, C2×C4 [×4], C2×C4 [×18], D4 [×16], C23 [×3], C23 [×4], C23 [×10], D5 [×4], C10, C10 [×2], C10 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×12], C24, C24, Dic5 [×4], Dic5 [×4], C20 [×4], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×D5 [×4], C2×Dic5 [×10], C2×Dic5 [×2], C5⋊D4 [×16], C2×C20 [×4], C2×C20 [×2], C22×D5 [×6], C22×D5 [×2], C22×C10 [×3], C22×C10 [×4], C22×C10 [×2], C22.11C24, C4×Dic5 [×4], C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×12], C22×C20 [×2], C23×D5, C23×C10, C23.11D10 [×2], D5×C22⋊C4 [×2], Dic54D4 [×4], C4×C5⋊D4 [×4], C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.24D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4 [×2], C4×D5 [×4], C22×D5 [×7], C22.11C24, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D46D10 [×2], C24.24D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A ··· 4H 4I ··· 4T 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 2 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 10 10 10 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 C4×D5 2+ 1+4 D4⋊6D10 kernel C24.24D10 C23.11D10 D5×C22⋊C4 Dic5⋊4D4 C4×C5⋊D4 C2×C23.D5 C10×C22⋊C4 C22×C5⋊D4 C2×C5⋊D4 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C10 C2 # reps 1 2 2 4 4 1 1 1 16 2 8 4 2 16 2 8

Matrix representation of C24.24D10 in GL6(𝔽41)

 40 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 14 2 40 0 0 0 14 2 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 40 0 0 0 0 36 23 0 0 0 0 9 0 17 40 0 0 28 32 1 24
,
 40 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 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
,
 22 22 0 0 0 0 19 32 0 0 0 0 0 0 31 22 27 33 0 0 9 13 35 34 0 0 8 6 19 19 0 0 8 13 19 19
,
 19 19 0 0 0 0 9 22 0 0 0 0 0 0 31 22 33 27 0 0 9 13 34 35 0 0 8 6 19 19 0 0 2 13 19 19

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,14,14,0,0,0,1,2,2,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,36,9,28,0,0,40,23,0,32,0,0,0,0,17,1,0,0,0,0,40,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[22,19,0,0,0,0,22,32,0,0,0,0,0,0,31,9,8,8,0,0,22,13,6,13,0,0,27,35,19,19,0,0,33,34,19,19],[19,9,0,0,0,0,19,22,0,0,0,0,0,0,31,9,8,2,0,0,22,13,6,13,0,0,33,34,19,19,0,0,27,35,19,19] >;`

C24.24D10 in GAP, Magma, Sage, TeX

`C_2^4._{24}D_{10}`
`% in TeX`

`G:=Group("C2^4.24D10");`
`// GroupNames label`

`G:=SmallGroup(320,1158);`
`// by ID`

`G=gap.SmallGroup(320,1158);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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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