Copied to
clipboard

## G = C40.C23order 320 = 26·5

### 6th non-split extension by C40 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40.C23
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — Q8.10D10 — C40.C23
 Lower central C5 — C10 — C20 — C40.C23
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for C40.C23
G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a29, cac-1=a31, dad-1=a11, bc=cb, dbd-1=a20b, cd=dc >

Subgroups: 1062 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, C22×D5, D4○SD16, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, Q8×D5, Q8×D5, Q82D5, Q82D5, Q8×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×SD16, D40⋊C2, Q16⋊D5, Q8.D10, C2×Q8⋊D5, D4.8D10, C5×C8.C22, Q8.10D10, D48D10, C40.C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○SD16, D4×D5, C23×D5, C2×D4×D5, C40.C23

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D4○SD16 D4×D5 D4×D5 C40.C23 kernel C40.C23 D20.2C4 C8⋊D10 D5×SD16 D40⋊C2 Q16⋊D5 Q8.D10 C2×Q8⋊D5 D4.8D10 C5×C8.C22 Q8.10D10 D4⋊8D10 Dic10 D20 C5⋊D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 4 4 2 2 2 2 2 2

Matrix representation of C40.C23 in GL8(𝔽41)

 0 0 34 7 0 0 0 0 0 0 34 1 0 0 0 0 7 34 0 0 0 0 0 0 7 40 0 0 0 0 0 0 0 0 0 0 0 15 0 15 0 0 0 0 0 0 15 15 0 0 0 0 30 0 26 15 0 0 0 0 30 11 26 15
,
 34 7 0 0 0 0 0 0 40 7 0 0 0 0 0 0 0 0 34 7 0 0 0 0 0 0 40 7 0 0 0 0 0 0 0 0 1 0 40 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 30 26 0 15 0 0 0 0 30 11 26 15 0 0 0 0 0 0 15 26 0 0 0 0 0 0 26 26
,
 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 11 15 0 26 0 0 0 0 0 0 26 26 0 0 0 0 11 0 15 26 0 0 0 0 30 11 26 15

`G:=sub<GL(8,GF(41))| [0,0,7,7,0,0,0,0,0,0,34,40,0,0,0,0,34,34,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,15,0,0,11,0,0,0,0,0,15,26,26,0,0,0,0,15,15,15,15],[34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,40,0,0,0,0,0,1,0,0,40],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,26,11,0,0,0,0,0,0,0,26,15,26,0,0,0,0,15,15,26,26],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,0,11,30,0,0,0,0,15,0,0,11,0,0,0,0,0,26,15,26,0,0,0,0,26,26,26,15] >;`

C40.C23 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,1450);`
`// by ID`

`G=gap.SmallGroup(320,1450);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=1,c^2=d^2=a^20,b*a*b=a^29,c*a*c^-1=a^31,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^20*b,c*d=d*c>;`
`// generators/relations`

׿
×
𝔽