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## G = C24.27D10order 320 = 26·5

### 27th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.27D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C24.27D10
 Lower central C5 — C2×C10 — C24.27D10
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C24.27D10
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, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1334 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×4], C22 [×25], C5, C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], D5 [×4], C10 [×3], C10 [×5], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×2], Dic5 [×4], C20 [×4], D10 [×2], D10 [×12], C2×C10, C2×C10 [×4], C2×C10 [×11], C2×C22⋊C4, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×5], D20 [×3], C2×Dic5 [×5], C2×Dic5 [×2], C5⋊D4 [×4], C5⋊D4 [×11], C2×C20 [×4], C2×C20 [×3], C22×D5 [×3], C22×D5 [×5], C22×C10 [×3], C22×C10 [×5], D45D4, C4×Dic5, C10.D4 [×3], C4⋊Dic5, D10⋊C4 [×5], C23.D5 [×3], C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×3], C2×D20 [×2], C4○D20 [×4], C22×Dic5, C2×C5⋊D4 [×7], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic54D4, C22⋊D20, D10.12D4, D10⋊D4 [×2], Dic5.5D4, C4×C5⋊D4, C23.23D10, C207D4, C242D5, C10×C22⋊C4, C2×C4○D20, C22×C5⋊D4, C24.27D10
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, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D46D10, C24.27D10

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 2 2 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 10 10 20 20 2 2 2 2 4 4 10 10 20 20 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 2 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 C4○D20 2+ 1+4 D4×D5 D4⋊6D10 kernel C24.27D10 Dic5.14D4 D5×C22⋊C4 Dic5⋊4D4 C22⋊D20 D10.12D4 D10⋊D4 Dic5.5D4 C4×C5⋊D4 C23.23D10 C20⋊7D4 C24⋊2D5 C10×C22⋊C4 C2×C4○D20 C22×C5⋊D4 C5⋊D4 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C22 C10 C22 C2 # reps 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 4 2 4 8 4 2 16 1 4 4

Matrix representation of C24.27D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 1 40
,
 23 6 0 0 35 18 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
,
 13 13 0 0 28 9 0 0 0 0 1 39 0 0 0 40
,
 13 13 0 0 9 28 0 0 0 0 1 39 0 0 0 40
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,1,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,40,0,0,0,0,40],[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],[13,28,0,0,13,9,0,0,0,0,1,0,0,0,39,40],[13,9,0,0,13,28,0,0,0,0,1,0,0,0,39,40] >;`

C24.27D10 in GAP, Magma, Sage, TeX

`C_2^4._{27}D_{10}`
`% in TeX`

`G:=Group("C2^4.27D10");`
`// GroupNames label`

`G:=SmallGroup(320,1162);`
`// by ID`

`G=gap.SmallGroup(320,1162);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,12550]);`
`// 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,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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