Copied to
clipboard

## G = C33⋊6D4order 216 = 23·33

### 3rd semidirect product of C33 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊6D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊6D4
 Lower central C33 — C32×C6 — C33⋊6D4
 Upper central C1 — C2

Generators and relations for C336D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 532 in 120 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×5], C6, C6 [×4], C6 [×9], D4, C32, C32 [×4], C32 [×4], Dic3 [×9], D6, D6 [×4], C2×C6 [×5], C3×S3 [×8], C3⋊S3, C3×C6, C3×C6 [×4], C3×C6 [×5], C3⋊D4 [×5], C33, C3⋊Dic3 [×9], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, D6⋊S3 [×4], C327D4, C335C4, S3×C3×C6, C6×C3⋊S3, C336D4
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], C3⋊S3, C3⋊D4 [×5], S32 [×4], C2×C3⋊S3, D6⋊S3 [×4], C327D4, S3×C3⋊S3, C336D4

Smallest permutation representation of C336D4
On 72 points
Generators in S72
```(1 54 63)(2 64 55)(3 56 61)(4 62 53)(5 16 59)(6 60 13)(7 14 57)(8 58 15)(9 23 40)(10 37 24)(11 21 38)(12 39 22)(17 71 30)(18 31 72)(19 69 32)(20 29 70)(25 52 34)(26 35 49)(27 50 36)(28 33 51)(41 46 65)(42 66 47)(43 48 67)(44 68 45)
(1 32 40)(2 37 29)(3 30 38)(4 39 31)(5 35 47)(6 48 36)(7 33 45)(8 46 34)(9 54 19)(10 20 55)(11 56 17)(12 18 53)(13 43 50)(14 51 44)(15 41 52)(16 49 42)(21 61 71)(22 72 62)(23 63 69)(24 70 64)(25 58 65)(26 66 59)(27 60 67)(28 68 57)
(1 23 19)(2 20 24)(3 21 17)(4 18 22)(5 49 66)(6 67 50)(7 51 68)(8 65 52)(9 32 63)(10 64 29)(11 30 61)(12 62 31)(13 48 27)(14 28 45)(15 46 25)(16 26 47)(33 44 57)(34 58 41)(35 42 59)(36 60 43)(37 55 70)(38 71 56)(39 53 72)(40 69 54)
(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 26)(2 25)(3 28)(4 27)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 45)(18 48)(19 47)(20 46)(29 65)(30 68)(31 67)(32 66)(33 56)(34 55)(35 54)(36 53)(37 58)(38 57)(39 60)(40 59)(41 70)(42 69)(43 72)(44 71)(49 63)(50 62)(51 61)(52 64)```

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

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

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

C336D4 is a maximal subgroup of
S3×D6⋊S3  D6⋊S32  D6.S32  D6.4S32  (C3×D12)⋊S3  C12.73S32  C12.57S32  C12⋊S32  C62.91D6  S3×C327D4  C3⋊S3×C3⋊D4
C336D4 is a maximal quotient of
C336D8  C3312SD16  C3313SD16  C336Q16  C62.77D6  C62.78D6  C62.81D6

33 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S order 1 2 2 2 3 ··· 3 3 3 3 3 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 size 1 1 6 18 2 ··· 2 4 4 4 4 54 2 ··· 2 4 4 4 4 6 ··· 6 18 18

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 S3 S3 D4 D6 C3⋊D4 S32 D6⋊S3 kernel C33⋊6D4 C33⋊5C4 S3×C3×C6 C6×C3⋊S3 S3×C6 C2×C3⋊S3 C33 C3×C6 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 10 4 4

Matrix representation of C336D4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 9 0 0 0 0 11 2 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 1 0
,
 11 9 0 0 0 0 4 2 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C336D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_6D_4`
`% in TeX`

`G:=Group("C3^3:6D4");`
`// GroupNames label`

`G:=SmallGroup(216,127);`
`// by ID`

`G=gap.SmallGroup(216,127);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,730,5189]);`
`// Polycyclic`

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

׿
×
𝔽