metabelian, supersoluble, monomial
Aliases: C33⋊7D8, C32⋊10D24, C12.26S32, (C3×D12)⋊3S3, D12⋊1(C3⋊S3), C32⋊4C8⋊8S3, (C3×C6).78D12, C3⋊2(C3⋊D24), C32⋊6(D4⋊S3), (C3×C12).113D6, (C32×D12)⋊4C2, C33⋊12D4⋊2C2, C3⋊1(C32⋊7D8), (C32×C6).30D4, C6.1(C32⋊7D4), C2.4(C33⋊7D4), C6.22(C3⋊D12), (C32×C12).9C22, C4.8(S3×C3⋊S3), C12.30(C2×C3⋊S3), (C3×C32⋊4C8)⋊2C2, (C3×C6).56(C3⋊D4), SmallGroup(432,437)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊7D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1672 in 196 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×14], C6, C6 [×4], C6 [×8], C8, D4 [×2], C32, C32 [×4], C32 [×4], C12, C12 [×4], C12 [×4], D6 [×14], C2×C6 [×4], D8, C3×S3 [×4], C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×5], C3⋊C8 [×4], C24, D12, D12 [×9], C3×D4 [×4], C33, C3×C12, C3×C12 [×4], C3×C12 [×4], S3×C6 [×4], C2×C3⋊S3 [×13], C62, D24, D4⋊S3 [×4], S3×C32, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C32⋊4C8, C3×D12 [×4], C12⋊S3 [×9], D4×C32, C32×C12, S3×C3×C6, C2×C33⋊C2, C3⋊D24 [×4], C32⋊7D8, C3×C32⋊4C8, C32×D12, C33⋊12D4, C33⋊7D8
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], D8, C3⋊S3, D12, C3⋊D4 [×4], S32 [×4], C2×C3⋊S3, D24, D4⋊S3 [×4], C3⋊D12 [×4], C32⋊7D4, S3×C3⋊S3, C3⋊D24 [×4], C32⋊7D8, C33⋊7D4, C33⋊7D8
►On 72 points
Generators in S72
(1 50 58)(2 59 51)(3 52 60)(4 61 53)(5 54 62)(6 63 55)(7 56 64)(8 57 49)(9 26 19)(10 20 27)(11 28 21)(12 22 29)(13 30 23)(14 24 31)(15 32 17)(16 18 25)(33 48 70)(34 71 41)(35 42 72)(36 65 43)(37 44 66)(38 67 45)(39 46 68)(40 69 47)
(1 25 47)(2 48 26)(3 27 41)(4 42 28)(5 29 43)(6 44 30)(7 31 45)(8 46 32)(9 51 33)(10 34 52)(11 53 35)(12 36 54)(13 55 37)(14 38 56)(15 49 39)(16 40 50)(17 57 68)(18 69 58)(19 59 70)(20 71 60)(21 61 72)(22 65 62)(23 63 66)(24 67 64)
(1 40 18)(2 33 19)(3 34 20)(4 35 21)(5 36 22)(6 37 23)(7 38 24)(8 39 17)(9 59 48)(10 60 41)(11 61 42)(12 62 43)(13 63 44)(14 64 45)(15 57 46)(16 58 47)(25 50 69)(26 51 70)(27 52 71)(28 53 72)(29 54 65)(30 55 66)(31 56 67)(32 49 68)
(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 7)(2 6)(3 5)(9 66)(10 65)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 40)(25 45)(26 44)(27 43)(28 42)(29 41)(30 48)(31 47)(32 46)(49 57)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)
G:=sub<Sym(72)| (1,50,58)(2,59,51)(3,52,60)(4,61,53)(5,54,62)(6,63,55)(7,56,64)(8,57,49)(9,26,19)(10,20,27)(11,28,21)(12,22,29)(13,30,23)(14,24,31)(15,32,17)(16,18,25)(33,48,70)(34,71,41)(35,42,72)(36,65,43)(37,44,66)(38,67,45)(39,46,68)(40,69,47), (1,25,47)(2,48,26)(3,27,41)(4,42,28)(5,29,43)(6,44,30)(7,31,45)(8,46,32)(9,51,33)(10,34,52)(11,53,35)(12,36,54)(13,55,37)(14,38,56)(15,49,39)(16,40,50)(17,57,68)(18,69,58)(19,59,70)(20,71,60)(21,61,72)(22,65,62)(23,63,66)(24,67,64), (1,40,18)(2,33,19)(3,34,20)(4,35,21)(5,36,22)(6,37,23)(7,38,24)(8,39,17)(9,59,48)(10,60,41)(11,61,42)(12,62,43)(13,63,44)(14,64,45)(15,57,46)(16,58,47)(25,50,69)(26,51,70)(27,52,71)(28,53,72)(29,54,65)(30,55,66)(31,56,67)(32,49,68), (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), (1,7)(2,6)(3,5)(9,66)(10,65)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,45)(26,44)(27,43)(28,42)(29,41)(30,48)(31,47)(32,46)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)>;
G:=Group( (1,50,58)(2,59,51)(3,52,60)(4,61,53)(5,54,62)(6,63,55)(7,56,64)(8,57,49)(9,26,19)(10,20,27)(11,28,21)(12,22,29)(13,30,23)(14,24,31)(15,32,17)(16,18,25)(33,48,70)(34,71,41)(35,42,72)(36,65,43)(37,44,66)(38,67,45)(39,46,68)(40,69,47), (1,25,47)(2,48,26)(3,27,41)(4,42,28)(5,29,43)(6,44,30)(7,31,45)(8,46,32)(9,51,33)(10,34,52)(11,53,35)(12,36,54)(13,55,37)(14,38,56)(15,49,39)(16,40,50)(17,57,68)(18,69,58)(19,59,70)(20,71,60)(21,61,72)(22,65,62)(23,63,66)(24,67,64), (1,40,18)(2,33,19)(3,34,20)(4,35,21)(5,36,22)(6,37,23)(7,38,24)(8,39,17)(9,59,48)(10,60,41)(11,61,42)(12,62,43)(13,63,44)(14,64,45)(15,57,46)(16,58,47)(25,50,69)(26,51,70)(27,52,71)(28,53,72)(29,54,65)(30,55,66)(31,56,67)(32,49,68), (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), (1,7)(2,6)(3,5)(9,66)(10,65)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,45)(26,44)(27,43)(28,42)(29,41)(30,48)(31,47)(32,46)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58) );
G=PermutationGroup([(1,50,58),(2,59,51),(3,52,60),(4,61,53),(5,54,62),(6,63,55),(7,56,64),(8,57,49),(9,26,19),(10,20,27),(11,28,21),(12,22,29),(13,30,23),(14,24,31),(15,32,17),(16,18,25),(33,48,70),(34,71,41),(35,42,72),(36,65,43),(37,44,66),(38,67,45),(39,46,68),(40,69,47)], [(1,25,47),(2,48,26),(3,27,41),(4,42,28),(5,29,43),(6,44,30),(7,31,45),(8,46,32),(9,51,33),(10,34,52),(11,53,35),(12,36,54),(13,55,37),(14,38,56),(15,49,39),(16,40,50),(17,57,68),(18,69,58),(19,59,70),(20,71,60),(21,61,72),(22,65,62),(23,63,66),(24,67,64)], [(1,40,18),(2,33,19),(3,34,20),(4,35,21),(5,36,22),(6,37,23),(7,38,24),(8,39,17),(9,59,48),(10,60,41),(11,61,42),(12,62,43),(13,63,44),(14,64,45),(15,57,46),(16,58,47),(25,50,69),(26,51,70),(27,52,71),(28,53,72),(29,54,65),(30,55,66),(31,56,67),(32,49,68)], [(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)], [(1,7),(2,6),(3,5),(9,66),(10,65),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,40),(25,45),(26,44),(27,43),(28,42),(29,41),(30,48),(31,47),(32,46),(49,57),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4 | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | ··· | 6Q | 8A | 8B | 12A | 12B | 12C | ··· | 12N | 24A | 24B | 24C | 24D |
| 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 | 24 | 24 |
| size | 1 | 1 | 12 | 108 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 |
51 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⋊7D8 | C3×C32⋊4C8 | C32×D12 | C33⋊12D4 | C32⋊4C8 | C3×D12 | C32×C6 | C3×C12 | C33 | C3×C6 | C3×C6 | C32 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 5 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C33⋊7D8 ►in GL8(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 1 | 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 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 29 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 33 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[12,8,0,0,0,0,0,0,16,29,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,33,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;
C33⋊7D8 in GAP, Magma, Sage, TeX
C_3^3\rtimes_7D_8
% in TeX
G:=Group("C3^3:7D8"); // GroupNames label
G:=SmallGroup(432,437);
// by ID
G=gap.SmallGroup(432,437);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,571,2028,14118]);
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations