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## G = C62.73D4order 288 = 25·32

### 57th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.73D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C62.73D4
 Lower central C32 — C3×C6 — C3×C12 — C62.73D4
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for C62.73D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=b3c3 >

Subgroups: 908 in 204 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×2], C4, C22, C22 [×5], S3 [×8], C6 [×4], C6 [×8], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C32, C12 [×8], C12 [×4], D6 [×16], C2×C6 [×4], C2×C6 [×4], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×8], D12 [×12], C2×C12 [×4], C2×C12 [×4], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×4], C22×S3 [×4], C8⋊C22, C3×C12 [×2], C3×C12, C2×C3⋊S3 [×4], C62, C62, C4.Dic3 [×4], D4⋊S3 [×8], Q82S3 [×8], C2×D12 [×4], C3×C4○D4 [×4], C324C8 [×2], C12⋊S3 [×2], C12⋊S3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, C22×C3⋊S3, D4⋊D6 [×4], C12.58D6, C327D8 [×2], C3211SD16 [×2], C2×C12⋊S3, C32×C4○D4, C62.73D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, D4⋊D6 [×4], C2×C327D4, C62.73D4

Smallest permutation representation of C62.73D4
On 72 points
Generators in S72
```(1 63 52 5 59 56)(2 53 60)(3 57 54 7 61 50)(4 55 62)(6 49 64)(8 51 58)(9 21 48)(10 45 22 14 41 18)(11 23 42)(12 47 24 16 43 20)(13 17 44)(15 19 46)(25 40 70 29 36 66)(26 71 37)(27 34 72 31 38 68)(28 65 39)(30 67 33)(32 69 35)
(1 27 20 5 31 24)(2 17 32 6 21 28)(3 29 22 7 25 18)(4 19 26 8 23 30)(9 39 60 13 35 64)(10 57 36 14 61 40)(11 33 62 15 37 58)(12 59 38 16 63 34)(41 50 70 45 54 66)(42 67 55 46 71 51)(43 52 72 47 56 68)(44 69 49 48 65 53)
(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)
(2 8)(3 7)(4 6)(9 67)(10 66)(11 65)(12 72)(13 71)(14 70)(15 69)(16 68)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E ··· 6P 8A 8B 12A ··· 12H 12I ··· 12T order 1 2 2 2 2 2 3 3 3 3 4 4 4 6 6 6 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 size 1 1 2 4 36 36 2 2 2 2 2 2 4 2 2 2 2 4 ··· 4 36 36 2 ··· 2 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 D4⋊D6 kernel C62.73D4 C12.58D6 C32⋊7D8 C32⋊11SD16 C2×C12⋊S3 C32×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 4 1 1 4 4 4 8 8 1 8

Matrix representation of C62.73D4 in GL6(𝔽73)

 1 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 12 57 1 0 0 0 28 12 0 1
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 62 39 59 0 0 0 50 11 0 14 0 0 64 72 11 39 0 0 72 8 50 62
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 12 28 66 66 0 0 28 16 59 7

`G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,12,28,0,0,0,72,57,12,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,62,50,64,72,0,0,39,11,72,8,0,0,59,0,11,50,0,0,0,14,39,62],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,12,28,0,0,72,1,28,16,0,0,0,0,66,59,0,0,0,0,66,7] >;`

C62.73D4 in GAP, Magma, Sage, TeX

`C_6^2._{73}D_4`
`% in TeX`

`G:=Group("C6^2.73D4");`
`// GroupNames label`

`G:=SmallGroup(288,806);`
`// by ID`

`G=gap.SmallGroup(288,806);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,675,185,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^3>;`
`// generators/relations`

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