Copied to
clipboard

## G = C36.48D4order 288 = 25·32

### 4th non-split extension by C36 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.48D4
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C2×D36 — C36.48D4
 Lower central C9 — C18 — C2×C18 — C36.48D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C36.48D4
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 436 in 69 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C9, C12, D6, C2×C6, M4(2), M4(2), C2×D4, D9, C18, C18, C3⋊C8, C24, D12, C2×C12, C22×S3, C4.D4, C36, D18, C2×C18, C4.Dic3, C3×M4(2), C2×D12, C9⋊C8, C72, D36, C2×C36, C22×D9, C12.46D4, C4.Dic9, C9×M4(2), C2×D36, C36.48D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.46D4, D18⋊C4, C36.48D4

Smallest permutation representation of C36.48D4
On 72 points
Generators in S72
```(1 59 23 50 14 68 32 41)(2 60 24 51 15 69 33 42)(3 61 25 52 16 70 34 43)(4 62 26 53 17 71 35 44)(5 63 27 54 18 72 36 45)(6 55 19 46 10 64 28 37)(7 56 20 47 11 65 29 38)(8 57 21 48 12 66 30 39)(9 58 22 49 13 67 31 40)
(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)
(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)
(1 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G 36H 36I 72A ··· 72L order 1 2 2 2 2 3 4 4 6 6 8 8 8 8 9 9 9 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 36 36 72 ··· 72 size 1 1 2 36 36 2 2 2 2 4 4 4 36 36 2 2 2 2 2 4 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 D9 D12 C3⋊D4 C4×S3 D18 D36 C9⋊D4 C4×D9 C4.D4 C12.46D4 C36.48D4 kernel C36.48D4 C4.Dic9 C9×M4(2) C2×D36 C22×D9 C3×M4(2) C36 C2×C12 M4(2) C12 C12 C2×C6 C2×C4 C4 C4 C22 C9 C3 C1 # reps 1 1 1 1 4 1 2 1 3 2 2 2 3 6 6 6 1 2 6

Matrix representation of C36.48D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 71 0 0 0 0 72 0 71 0 0 4 66 1 0 0 0 7 70 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 72 0 72 0 0 0 0 72 0 72
,
 16 0 0 0 0 0 0 32 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 0 32 0 0 0 0 16 0 0 0 0 0 0 0 66 14 0 0 0 0 7 7 0 0 0 0 0 0 66 14 0 0 0 0 7 7

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,4,7,0,0,0,72,66,70,0,0,71,0,1,0,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,72,0,0,0,0,72,0,0,0,0,0,0,72],[16,0,0,0,0,0,0,32,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,16,0,0,0,0,32,0,0,0,0,0,0,0,66,7,0,0,0,0,14,7,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;`

C36.48D4 in GAP, Magma, Sage, TeX

`C_{36}._{48}D_4`
`% in TeX`

`G:=Group("C36.48D4");`
`// GroupNames label`

`G:=SmallGroup(288,31);`
`// by ID`

`G=gap.SmallGroup(288,31);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽