metabelian, supersoluble, monomial
Aliases: C33⋊8D8, C32⋊5D24, C12.52S32, C12⋊S3⋊7S3, (C3×C6).37D12, C3⋊2(C32⋊5D8), C3⋊1(C3⋊D24), C32⋊9(D4⋊S3), (C3×C12).114D6, C33⋊12D4⋊3C2, (C32×C6).31D4, C6.8(C3⋊D12), C6.12(C12⋊S3), C2.4(C33⋊8D4), (C32×C12).10C22, (C3×C3⋊C8)⋊1S3, C3⋊C8⋊1(C3⋊S3), C4.1(S3×C3⋊S3), (C32×C3⋊C8)⋊1C2, C12.10(C2×C3⋊S3), (C3×C12⋊S3)⋊3C2, (C3×C6).77(C3⋊D4), SmallGroup(432,438)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊8D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1792 in 196 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C3×C3⋊C8, C3×C24, C3×D12, C12⋊S3, C12⋊S3, C32×C12, C6×C3⋊S3, C2×C33⋊C2, C3⋊D24, C32⋊5D8, C32×C3⋊C8, C3×C12⋊S3, C33⋊12D4, C33⋊8D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, D24, D4⋊S3, C3⋊D12, C12⋊S3, S3×C3⋊S3, C3⋊D24, C32⋊5D8, C33⋊8D4, C33⋊8D8
►On 72 points
Generators in S72
(1 38 24)(2 39 17)(3 40 18)(4 33 19)(5 34 20)(6 35 21)(7 36 22)(8 37 23)(9 67 52)(10 68 53)(11 69 54)(12 70 55)(13 71 56)(14 72 49)(15 65 50)(16 66 51)(25 41 58)(26 42 59)(27 43 60)(28 44 61)(29 45 62)(30 46 63)(31 47 64)(32 48 57)
(1 25 69)(2 26 70)(3 27 71)(4 28 72)(5 29 65)(6 30 66)(7 31 67)(8 32 68)(9 22 64)(10 23 57)(11 24 58)(12 17 59)(13 18 60)(14 19 61)(15 20 62)(16 21 63)(33 44 49)(34 45 50)(35 46 51)(36 47 52)(37 48 53)(38 41 54)(39 42 55)(40 43 56)
(1 11 41)(2 42 12)(3 13 43)(4 44 14)(5 15 45)(6 46 16)(7 9 47)(8 48 10)(17 26 55)(18 56 27)(19 28 49)(20 50 29)(21 30 51)(22 52 31)(23 32 53)(24 54 25)(33 61 72)(34 65 62)(35 63 66)(36 67 64)(37 57 68)(38 69 58)(39 59 70)(40 71 60)
(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 43)(10 42)(11 41)(12 48)(13 47)(14 46)(15 45)(16 44)(17 37)(18 36)(19 35)(20 34)(21 33)(22 40)(23 39)(24 38)(25 69)(26 68)(27 67)(28 66)(29 65)(30 72)(31 71)(32 70)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 64)
G:=sub<Sym(72)| (1,38,24)(2,39,17)(3,40,18)(4,33,19)(5,34,20)(6,35,21)(7,36,22)(8,37,23)(9,67,52)(10,68,53)(11,69,54)(12,70,55)(13,71,56)(14,72,49)(15,65,50)(16,66,51)(25,41,58)(26,42,59)(27,43,60)(28,44,61)(29,45,62)(30,46,63)(31,47,64)(32,48,57), (1,25,69)(2,26,70)(3,27,71)(4,28,72)(5,29,65)(6,30,66)(7,31,67)(8,32,68)(9,22,64)(10,23,57)(11,24,58)(12,17,59)(13,18,60)(14,19,61)(15,20,62)(16,21,63)(33,44,49)(34,45,50)(35,46,51)(36,47,52)(37,48,53)(38,41,54)(39,42,55)(40,43,56), (1,11,41)(2,42,12)(3,13,43)(4,44,14)(5,15,45)(6,46,16)(7,9,47)(8,48,10)(17,26,55)(18,56,27)(19,28,49)(20,50,29)(21,30,51)(22,52,31)(23,32,53)(24,54,25)(33,61,72)(34,65,62)(35,63,66)(36,67,64)(37,57,68)(38,69,58)(39,59,70)(40,71,60), (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)(9,43)(10,42)(11,41)(12,48)(13,47)(14,46)(15,45)(16,44)(17,37)(18,36)(19,35)(20,34)(21,33)(22,40)(23,39)(24,38)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,64)>;
G:=Group( (1,38,24)(2,39,17)(3,40,18)(4,33,19)(5,34,20)(6,35,21)(7,36,22)(8,37,23)(9,67,52)(10,68,53)(11,69,54)(12,70,55)(13,71,56)(14,72,49)(15,65,50)(16,66,51)(25,41,58)(26,42,59)(27,43,60)(28,44,61)(29,45,62)(30,46,63)(31,47,64)(32,48,57), (1,25,69)(2,26,70)(3,27,71)(4,28,72)(5,29,65)(6,30,66)(7,31,67)(8,32,68)(9,22,64)(10,23,57)(11,24,58)(12,17,59)(13,18,60)(14,19,61)(15,20,62)(16,21,63)(33,44,49)(34,45,50)(35,46,51)(36,47,52)(37,48,53)(38,41,54)(39,42,55)(40,43,56), (1,11,41)(2,42,12)(3,13,43)(4,44,14)(5,15,45)(6,46,16)(7,9,47)(8,48,10)(17,26,55)(18,56,27)(19,28,49)(20,50,29)(21,30,51)(22,52,31)(23,32,53)(24,54,25)(33,61,72)(34,65,62)(35,63,66)(36,67,64)(37,57,68)(38,69,58)(39,59,70)(40,71,60), (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)(9,43)(10,42)(11,41)(12,48)(13,47)(14,46)(15,45)(16,44)(17,37)(18,36)(19,35)(20,34)(21,33)(22,40)(23,39)(24,38)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,64) );
G=PermutationGroup([[(1,38,24),(2,39,17),(3,40,18),(4,33,19),(5,34,20),(6,35,21),(7,36,22),(8,37,23),(9,67,52),(10,68,53),(11,69,54),(12,70,55),(13,71,56),(14,72,49),(15,65,50),(16,66,51),(25,41,58),(26,42,59),(27,43,60),(28,44,61),(29,45,62),(30,46,63),(31,47,64),(32,48,57)], [(1,25,69),(2,26,70),(3,27,71),(4,28,72),(5,29,65),(6,30,66),(7,31,67),(8,32,68),(9,22,64),(10,23,57),(11,24,58),(12,17,59),(13,18,60),(14,19,61),(15,20,62),(16,21,63),(33,44,49),(34,45,50),(35,46,51),(36,47,52),(37,48,53),(38,41,54),(39,42,55),(40,43,56)], [(1,11,41),(2,42,12),(3,13,43),(4,44,14),(5,15,45),(6,46,16),(7,9,47),(8,48,10),(17,26,55),(18,56,27),(19,28,49),(20,50,29),(21,30,51),(22,52,31),(23,32,53),(24,54,25),(33,61,72),(34,65,62),(35,63,66),(36,67,64),(37,57,68),(38,69,58),(39,59,70),(40,71,60)], [(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),(9,43),(10,42),(11,41),(12,48),(13,47),(14,46),(15,45),(16,44),(17,37),(18,36),(19,35),(20,34),(21,33),(22,40),(23,39),(24,38),(25,69),(26,68),(27,67),(28,66),(29,65),(30,72),(31,71),(32,70),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,64)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4 | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12Q | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 108 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 36 | 36 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D8 | D12 | C3⋊D4 | D24 | S32 | D4⋊S3 | C3⋊D12 | C3⋊D24 |
| kernel | C33⋊8D8 | C32×C3⋊C8 | C3×C12⋊S3 | C33⋊12D4 | C3×C3⋊C8 | C12⋊S3 | C32×C6 | C3×C12 | C33 | C3×C6 | C3×C6 | C32 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 2 | 8 | 2 | 16 | 4 | 1 | 4 | 8 |
Matrix representation of C33⋊8D8 ►in GL6(𝔽73)
| 1 | 70 | 0 | 0 | 0 | 0 |
| 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 71 | 3 | 0 | 0 | 0 | 0 |
| 72 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 15 | 0 | 0 | 0 | 0 |
| 68 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 72 | 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,70,71,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[71,72,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[13,68,0,0,0,0,15,28,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C33⋊8D8 in GAP, Magma, Sage, TeX
C_3^3\rtimes_8D_8
% in TeX
G:=Group("C3^3:8D8"); // GroupNames label
G:=SmallGroup(432,438);
// by ID
G=gap.SmallGroup(432,438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,85,135,58,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations