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## G = C62.38D4order 288 = 25·32

### 22nd non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.38D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — C62⋊5C4 — C62.38D4
 Lower central C32 — C3×C6 — C62 — C62.38D4
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C62.38D4
G = < a,b,c,d | a6=b6=c4=1, d2=a3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 540 in 156 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C2×D4, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×D4, C22×C6, C23⋊C4, C3⋊Dic3, C3×C12, C62, C62, C62, C6.D4, C6×D4, C2×C3⋊Dic3, C6×C12, D4×C32, C2×C62, C23.7D6, C625C4, D4×C3×C6, C62.38D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C23⋊C4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C327D4, C23.7D6, C625C4, C62.38D4

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

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 4 2 2 2 2 4 36 36 36 36 2 ··· 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 S3 D4 Dic3 Dic3 D6 C3⋊D4 C23⋊C4 C23.7D6 kernel C62.38D4 C62⋊5C4 D4×C3×C6 C6×C12 C2×C62 C6×D4 C62 C2×C12 C22×C6 C22×C6 C2×C6 C32 C3 # reps 1 2 1 2 2 4 2 4 4 4 16 1 8

Matrix representation of C62.38D4 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 9 0 12
,
 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 2 8 0 4 0 0 0 0 5 7 9 10 0 0 0 0 1 0 0 0 0 0 0 0 5 12 8 4
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 2 8 0 4 0 0 0 0 9 2 3 0

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,1,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,2,5,1,5,0,0,0,0,8,7,0,12,0,0,0,0,0,9,0,8,0,0,0,0,4,10,0,4],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,5,2,9,0,0,0,0,0,12,8,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0] >;`

C62.38D4 in GAP, Magma, Sage, TeX

`C_6^2._{38}D_4`
`% in TeX`

`G:=Group("C6^2.38D4");`
`// GroupNames label`

`G:=SmallGroup(288,309);`
`// by ID`

`G=gap.SmallGroup(288,309);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,675,2693,9414]);`
`// Polycyclic`

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

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