Copied to
clipboard

## G = C40.47C23order 320 = 26·5

### 40th non-split extension by C40 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C40.47C23
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×C4○D20 — C40.47C23
 Lower central C5 — C10 — C40.47C23
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C40.47C23
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a5, bab=a9, ac=ca, ad=da, ae=ea, bc=cb, dbd=a10b, be=eb, cd=dc, ece-1=a10c, de=ed >

Subgroups: 718 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Q8○M4(2), C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D20.3C4, D5×M4(2), D20.2C4, C2×C4.Dic5, C10×M4(2), C2×C4○D20, C40.47C23
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, C4×D5, C22×D5, Q8○M4(2), C2×C4×D5, C23×D5, D5×C22×C4, C40.47C23

Smallest permutation representation of C40.47C23
On 80 points
Generators in S80
```(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)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(22 30)(23 39)(24 28)(25 37)(27 35)(29 33)(32 40)(34 38)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 31 6 36 11 21 16 26)(2 32 7 37 12 22 17 27)(3 33 8 38 13 23 18 28)(4 34 9 39 14 24 19 29)(5 35 10 40 15 25 20 30)(41 66 46 71 51 76 56 61)(42 67 47 72 52 77 57 62)(43 68 48 73 53 78 58 63)(44 69 49 74 54 79 59 64)(45 70 50 75 55 80 60 65)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 2 2 10 10 10 10 1 1 2 2 2 10 10 10 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 D10 D10 D10 C4×D5 C4×D5 Q8○M4(2) C40.47C23 kernel C40.47C23 D20.3C4 D5×M4(2) D20.2C4 C2×C4.Dic5 C10×M4(2) C2×C4○D20 C2×Dic10 C2×D20 C4○D20 C2×C5⋊D4 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C5 C1 # reps 1 4 4 4 1 1 1 2 2 8 4 2 4 8 2 12 4 2 8

Matrix representation of C40.47C23 in GL4(𝔽41) generated by

 19 32 0 0 9 0 0 0 0 0 19 32 0 0 9 0
,
 1 0 0 0 34 40 0 0 0 0 1 0 0 0 34 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 17 40 0 0 1 24 0 0 0 0 17 40 0 0 1 24
,
 0 0 1 0 0 0 0 1 9 0 0 0 0 9 0 0
`G:=sub<GL(4,GF(41))| [19,9,0,0,32,0,0,0,0,0,19,9,0,0,32,0],[1,34,0,0,0,40,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[17,1,0,0,40,24,0,0,0,0,17,1,0,0,40,24],[0,0,9,0,0,0,0,9,1,0,0,0,0,1,0,0] >;`

C40.47C23 in GAP, Magma, Sage, TeX

`C_{40}._{47}C_2^3`
`% in TeX`

`G:=Group("C40.47C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1417);`
`// by ID`

`G=gap.SmallGroup(320,1417);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^5,b*a*b=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=a^10*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^10*c,d*e=e*d>;`
`// generators/relations`

׿
×
𝔽