Copied to
clipboard

## G = C12.58D6order 144 = 24·32

### 19th non-split extension by C12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.58D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C12.58D6
 Lower central C32 — C3×C6 — C12.58D6
 Upper central C1 — C4 — C2×C4

Generators and relations for C12.58D6
G = < a,b,c | a12=b6=1, c2=a3, ab=ba, cac-1=a5, cbc-1=a6b-1 >

Subgroups: 114 in 60 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], M4(2), C3×C6, C3×C6, C3⋊C8 [×8], C2×C12 [×4], C3×C12 [×2], C62, C4.Dic3 [×4], C324C8 [×2], C6×C12, C12.58D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, Dic3 [×8], D6 [×4], M4(2), C3⋊S3, C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4.Dic3 [×4], C2×C3⋊Dic3, C12.58D6

Smallest permutation representation of C12.58D6
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 40 23)(2 41 24)(3 42 13)(4 43 14)(5 44 15)(6 45 16)(7 46 17)(8 47 18)(9 48 19)(10 37 20)(11 38 21)(12 39 22)(25 64 55 31 70 49)(26 65 56 32 71 50)(27 66 57 33 72 51)(28 67 58 34 61 52)(29 68 59 35 62 53)(30 69 60 36 63 54)
(1 63 4 66 7 69 10 72)(2 68 5 71 8 62 11 65)(3 61 6 64 9 67 12 70)(13 58 16 49 19 52 22 55)(14 51 17 54 20 57 23 60)(15 56 18 59 21 50 24 53)(25 42 28 45 31 48 34 39)(26 47 29 38 32 41 35 44)(27 40 30 43 33 46 36 37)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P order 1 2 2 3 3 3 3 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 2 2 2 2 2 1 1 2 2 ··· 2 18 18 18 18 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 kernel C12.58D6 C32⋊4C8 C6×C12 C3×C12 C62 C2×C12 C12 C12 C2×C6 C32 C3 # reps 1 2 1 2 2 4 4 4 4 2 16

Matrix representation of C12.58D6 in GL4(𝔽73) generated by

 49 0 0 0 0 70 0 0 0 0 3 0 0 0 0 24
,
 1 0 0 0 0 72 0 0 0 0 64 0 0 0 0 65
,
 0 1 0 0 46 0 0 0 0 0 0 8 0 0 49 0
`G:=sub<GL(4,GF(73))| [49,0,0,0,0,70,0,0,0,0,3,0,0,0,0,24],[1,0,0,0,0,72,0,0,0,0,64,0,0,0,0,65],[0,46,0,0,1,0,0,0,0,0,0,49,0,0,8,0] >;`

C12.58D6 in GAP, Magma, Sage, TeX

`C_{12}._{58}D_6`
`% in TeX`

`G:=Group("C12.58D6");`
`// GroupNames label`

`G:=SmallGroup(144,91);`
`// by ID`

`G=gap.SmallGroup(144,91);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,50,964,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^6=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;`
`// generators/relations`

׿
×
𝔽