direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32xC3:C8, C33:5C8, C32:5C24, C3:(C3xC24), C6.(C3xC12), C12.2(C3xC6), (C3xC6).8C12, C12.20(C3xS3), (C3xC12).22S3, (C3xC12).15C6, (C32xC6).3C4, C4.2(S3xC32), C2.(C32xDic3), (C32xC12).4C2, (C3xC6).12Dic3, C6.10(C3xDic3), SmallGroup(216,82)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C32xC3:C8 |
Generators and relations for C32xC3:C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 108 in 72 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C3xC6, C3xC6, C3xC6, C3:C8, C24, C33, C3xC12, C3xC12, C3xC12, C32xC6, C3xC3:C8, C3xC24, C32xC12, C32xC3:C8
Quotients: C1, C2, C3, C4, S3, C6, C8, C32, Dic3, C12, C3xS3, C3xC6, C3:C8, C24, C3xDic3, C3xC12, S3xC32, C3xC3:C8, C3xC24, C32xDic3, C32xC3:C8
►On 72 points
Generators in S72
(1 45 13)(2 46 14)(3 47 15)(4 48 16)(5 41 9)(6 42 10)(7 43 11)(8 44 12)(17 72 58)(18 65 59)(19 66 60)(20 67 61)(21 68 62)(22 69 63)(23 70 64)(24 71 57)(25 53 39)(26 54 40)(27 55 33)(28 56 34)(29 49 35)(30 50 36)(31 51 37)(32 52 38)
(1 51 71)(2 52 72)(3 53 65)(4 54 66)(5 55 67)(6 56 68)(7 49 69)(8 50 70)(9 27 20)(10 28 21)(11 29 22)(12 30 23)(13 31 24)(14 32 17)(15 25 18)(16 26 19)(33 61 41)(34 62 42)(35 63 43)(36 64 44)(37 57 45)(38 58 46)(39 59 47)(40 60 48)
(1 51 71)(2 72 52)(3 53 65)(4 66 54)(5 55 67)(6 68 56)(7 49 69)(8 70 50)(9 27 20)(10 21 28)(11 29 22)(12 23 30)(13 31 24)(14 17 32)(15 25 18)(16 19 26)(33 61 41)(34 42 62)(35 63 43)(36 44 64)(37 57 45)(38 46 58)(39 59 47)(40 48 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)
G:=sub<Sym(72)| (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,72,58)(18,65,59)(19,66,60)(20,67,61)(21,68,62)(22,69,63)(23,70,64)(24,71,57)(25,53,39)(26,54,40)(27,55,33)(28,56,34)(29,49,35)(30,50,36)(31,51,37)(32,52,38), (1,51,71)(2,52,72)(3,53,65)(4,54,66)(5,55,67)(6,56,68)(7,49,69)(8,50,70)(9,27,20)(10,28,21)(11,29,22)(12,30,23)(13,31,24)(14,32,17)(15,25,18)(16,26,19)(33,61,41)(34,62,42)(35,63,43)(36,64,44)(37,57,45)(38,58,46)(39,59,47)(40,60,48), (1,51,71)(2,72,52)(3,53,65)(4,66,54)(5,55,67)(6,68,56)(7,49,69)(8,70,50)(9,27,20)(10,21,28)(11,29,22)(12,23,30)(13,31,24)(14,17,32)(15,25,18)(16,19,26)(33,61,41)(34,42,62)(35,63,43)(36,44,64)(37,57,45)(38,46,58)(39,59,47)(40,48,60), (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)>;
G:=Group( (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,72,58)(18,65,59)(19,66,60)(20,67,61)(21,68,62)(22,69,63)(23,70,64)(24,71,57)(25,53,39)(26,54,40)(27,55,33)(28,56,34)(29,49,35)(30,50,36)(31,51,37)(32,52,38), (1,51,71)(2,52,72)(3,53,65)(4,54,66)(5,55,67)(6,56,68)(7,49,69)(8,50,70)(9,27,20)(10,28,21)(11,29,22)(12,30,23)(13,31,24)(14,32,17)(15,25,18)(16,26,19)(33,61,41)(34,62,42)(35,63,43)(36,64,44)(37,57,45)(38,58,46)(39,59,47)(40,60,48), (1,51,71)(2,72,52)(3,53,65)(4,66,54)(5,55,67)(6,68,56)(7,49,69)(8,70,50)(9,27,20)(10,21,28)(11,29,22)(12,23,30)(13,31,24)(14,17,32)(15,25,18)(16,19,26)(33,61,41)(34,42,62)(35,63,43)(36,44,64)(37,57,45)(38,46,58)(39,59,47)(40,48,60), (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) );
G=PermutationGroup([[(1,45,13),(2,46,14),(3,47,15),(4,48,16),(5,41,9),(6,42,10),(7,43,11),(8,44,12),(17,72,58),(18,65,59),(19,66,60),(20,67,61),(21,68,62),(22,69,63),(23,70,64),(24,71,57),(25,53,39),(26,54,40),(27,55,33),(28,56,34),(29,49,35),(30,50,36),(31,51,37),(32,52,38)], [(1,51,71),(2,52,72),(3,53,65),(4,54,66),(5,55,67),(6,56,68),(7,49,69),(8,50,70),(9,27,20),(10,28,21),(11,29,22),(12,30,23),(13,31,24),(14,32,17),(15,25,18),(16,26,19),(33,61,41),(34,62,42),(35,63,43),(36,64,44),(37,57,45),(38,58,46),(39,59,47),(40,60,48)], [(1,51,71),(2,72,52),(3,53,65),(4,66,54),(5,55,67),(6,68,56),(7,49,69),(8,70,50),(9,27,20),(10,21,28),(11,29,22),(12,23,30),(13,31,24),(14,17,32),(15,25,18),(16,19,26),(33,61,41),(34,42,62),(35,63,43),(36,44,64),(37,57,45),(38,46,58),(39,59,47),(40,48,60)], [(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)]])
C32xC3:C8 is a maximal subgroup of
C12.69S32 C33:8M4(2) C33:9M4(2) C33:8D8 C33:16SD16 C33:17SD16 C33:8Q16 S3xC3xC24
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AH | 24A | ··· | 24AF |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3xS3 | C3:C8 | C3xDic3 | C3xC3:C8 |
| kernel | C32xC3:C8 | C32xC12 | C3xC3:C8 | C32xC6 | C3xC12 | C33 | C3xC6 | C32 | C3xC12 | C3xC6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 32 | 1 | 1 | 8 | 2 | 8 | 16 |
Matrix representation of C32xC3:C8 ►in GL3(F73) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 22 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[64,0,0,0,8,0,0,0,8],[1,0,0,0,8,0,0,0,64],[22,0,0,0,0,27,0,1,0] >;
C32xC3:C8 in GAP, Magma, Sage, TeX
C_3^2\times C_3\rtimes C_8
% in TeX
G:=Group("C3^2xC3:C8"); // GroupNames label
G:=SmallGroup(216,82);
// by ID
G=gap.SmallGroup(216,82);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-2,-3,108,69,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations