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## G = C24.42D10order 320 = 26·5

### 42nd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.42D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×D4⋊2D5 — C24.42D10
 Lower central C5 — C2×C10 — C24.42D10
 Upper central C1 — C22 — C22×D4

Generators and relations for C24.42D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1078 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C22×C10, C22×C10, D45D4, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, D4×C10, C23×C10, C20.48D4, C4×C5⋊D4, D4×Dic5, C23.18D10, C20.17D4, C202D4, Dic5⋊D4, C2×C23.D5, C242D5, C2×D42D5, D4×C2×C10, C24.42D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C5⋊D4, C22×D5, D45D4, D42D5, C2×C5⋊D4, C23×D5, C2×D42D5, D46D10, C22×C5⋊D4, C24.42D10

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 ··· 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 20 2 2 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 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 D4⋊2D5 D4⋊6D10 kernel C24.42D10 C20.48D4 C4×C5⋊D4 D4×Dic5 C23.18D10 C20.17D4 C20⋊2D4 Dic5⋊D4 C2×C23.D5 C24⋊2D5 C2×D4⋊2D5 D4×C2×C10 C5×D4 C22×D4 C2×C10 C22×C4 C2×D4 C24 D4 C10 C22 C2 # reps 1 1 1 1 2 1 1 2 2 2 1 1 4 2 4 2 8 4 16 1 4 4

Matrix representation of C24.42D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 40
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 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
,
 37 0 0 0 0 10 0 0 0 0 1 39 0 0 1 40
,
 0 31 0 0 4 0 0 0 0 0 9 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[37,0,0,0,0,10,0,0,0,0,1,1,0,0,39,40],[0,4,0,0,31,0,0,0,0,0,9,0,0,0,0,9] >;`

C24.42D10 in GAP, Magma, Sage, TeX

`C_2^4._{42}D_{10}`
`% in TeX`

`G:=Group("C2^4.42D10");`
`// GroupNames label`

`G:=SmallGroup(320,1478);`
`// by ID`

`G=gap.SmallGroup(320,1478);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,12550]);`
`// 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,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,f*b*f^-1=b*c=c*b,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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