direct product, metabelian, supersoluble, monomial
Aliases: C2xC32:5SD16, Dic6:17D6, C12.24D12, C62.51D4, C3:C8:25D6, (C3xC6):5SD16, C6:2(C24:C2), (C2xDic6):1S3, (C6xDic6):3C2, (C2xC6).61D12, (C3xC12).69D4, C6.71(C2xD12), C32:8(C2xSD16), C6:1(Q8:2S3), (C2xC12).121D6, C4.7(C3:D12), C12.72(C3:D4), (C6xC12).81C22, (C3xC12).68C23, C12.127(C22xS3), (C3xDic6):21C22, C12:S3.27C22, C22.21(C3:D12), (C2xC3:C8):8S3, (C6xC3:C8):12C2, C4.56(C2xS32), (C2xC4).67S32, C3:3(C2xC24:C2), C6.7(C2xC3:D4), (C3xC3:C8):30C22, C3:1(C2xQ8:2S3), (C3xC6).72(C2xD4), C2.11(C2xC3:D12), (C2xC6).39(C3:D4), (C2xC12:S3).11C2, SmallGroup(288,480)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC32:5SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d3 >
Subgroups: 834 in 163 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, Dic6, D12, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C2xSD16, C3xDic3, C3xC12, C2xC3:S3, C62, C24:C2, C2xC3:C8, Q8:2S3, C2xC24, C2xDic6, C2xD12, C6xQ8, C3xC3:C8, C3xDic6, C3xDic6, C6xDic3, C12:S3, C12:S3, C6xC12, C22xC3:S3, C2xC24:C2, C2xQ8:2S3, C32:5SD16, C6xC3:C8, C6xDic6, C2xC12:S3, C2xC32:5SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, D12, C3:D4, C22xS3, C2xSD16, S32, C24:C2, Q8:2S3, C2xD12, C2xC3:D4, C3:D12, C2xS32, C2xC24:C2, C2xQ8:2S3, C32:5SD16, C2xC3:D12, C2xC32:5SD16
►On 48 points
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 21 35)(2 36 22)(3 23 37)(4 38 24)(5 17 39)(6 40 18)(7 19 33)(8 34 20)(9 25 44)(10 45 26)(11 27 46)(12 47 28)(13 29 48)(14 41 30)(15 31 42)(16 43 32)
(1 35 21)(2 36 22)(3 37 23)(4 38 24)(5 39 17)(6 40 18)(7 33 19)(8 34 20)(9 44 25)(10 45 26)(11 46 27)(12 47 28)(13 48 29)(14 41 30)(15 42 31)(16 43 32)
(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)
(1 3)(2 6)(5 7)(9 11)(10 14)(13 15)(17 33)(18 36)(19 39)(20 34)(21 37)(22 40)(23 35)(24 38)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,21,35)(2,36,22)(3,23,37)(4,38,24)(5,17,39)(6,40,18)(7,19,33)(8,34,20)(9,25,44)(10,45,26)(11,27,46)(12,47,28)(13,29,48)(14,41,30)(15,31,42)(16,43,32), (1,35,21)(2,36,22)(3,37,23)(4,38,24)(5,39,17)(6,40,18)(7,33,19)(8,34,20)(9,44,25)(10,45,26)(11,46,27)(12,47,28)(13,48,29)(14,41,30)(15,42,31)(16,43,32), (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,21,35)(2,36,22)(3,23,37)(4,38,24)(5,17,39)(6,40,18)(7,19,33)(8,34,20)(9,25,44)(10,45,26)(11,27,46)(12,47,28)(13,29,48)(14,41,30)(15,31,42)(16,43,32), (1,35,21)(2,36,22)(3,37,23)(4,38,24)(5,39,17)(6,40,18)(7,33,19)(8,34,20)(9,44,25)(10,45,26)(11,46,27)(12,47,28)(13,48,29)(14,41,30)(15,42,31)(16,43,32), (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,21,35),(2,36,22),(3,23,37),(4,38,24),(5,17,39),(6,40,18),(7,19,33),(8,34,20),(9,25,44),(10,45,26),(11,27,46),(12,47,28),(13,29,48),(14,41,30),(15,31,42),(16,43,32)], [(1,35,21),(2,36,22),(3,37,23),(4,38,24),(5,39,17),(6,40,18),(7,33,19),(8,34,20),(9,44,25),(10,45,26),(11,46,27),(12,47,28),(13,48,29),(14,41,30),(15,42,31),(16,43,32)], [(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)], [(1,3),(2,6),(5,7),(9,11),(10,14),(13,15),(17,33),(18,36),(19,39),(20,34),(21,37),(22,40),(23,35),(24,38),(25,46),(26,41),(27,44),(28,47),(29,42),(30,45),(31,48),(32,43)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | D12 | C3:D4 | D12 | C3:D4 | C24:C2 | S32 | Q8:2S3 | C3:D12 | C2xS32 | C3:D12 | C32:5SD16 |
| kernel | C2xC32:5SD16 | C32:5SD16 | C6xC3:C8 | C6xDic6 | C2xC12:S3 | C2xC3:C8 | C2xDic6 | C3xC12 | C62 | C3:C8 | Dic6 | C2xC12 | C3xC6 | C12 | C12 | C2xC6 | C2xC6 | C6 | C2xC4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 4 |
Matrix representation of C2xC32:5SD16 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2xC32:5SD16 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_5{\rm SD}_{16} % in TeX
G:=Group("C2xC3^2:5SD16"); // GroupNames label
G:=SmallGroup(288,480);
// by ID
G=gap.SmallGroup(288,480);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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^3>;
// generators/relations