direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3xC22:SD16, C22:C8:9C6, D4.6(C3xD4), C4.24(C6xD4), C22:Q8:1C6, D4:C4:9C6, (C2xSD16):9C6, (C3xD4).40D4, (C2xC6):10SD16, C2.6(C6xSD16), (C2xC24):34C22, C6.97C22wrC2, (C6xSD16):26C2, C12.385(C2xD4), (C2xC12).319D4, C6.86(C2xSD16), (C6xQ8):26C22, (C22xD4).9C6, C22:3(C3xSD16), C23.48(C3xD4), C22.80(C6xD4), (C22xC6).165D4, C6.133(C8:C22), (C2xC12).915C23, (C6xD4).295C22, (C22xC12).422C22, C4:C4:2(C2xC6), (C2xC8):6(C2xC6), (C2xQ8):2(C2xC6), (D4xC2xC6).20C2, (C2xC4).28(C3xD4), C2.8(C3xC8:C22), (C3xC22:C8):26C2, (C3xC4:C4):36C22, (C2xD4).53(C2xC6), (C2xC6).636(C2xD4), (C3xC22:Q8):28C2, (C3xD4:C4):33C2, C2.11(C3xC22wrC2), (C22xC4).45(C2xC6), (C2xC4).90(C22xC6), SmallGroup(192,883)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC22:SD16
G = < a,b,c,d,e | a3=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 402 in 188 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xC6, C22xC6, C22:C8, D4:C4, C22:Q8, C2xSD16, C22xD4, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xC24, C3xSD16, C22xC12, C6xD4, C6xD4, C6xQ8, C23xC6, C22:SD16, C3xC22:C8, C3xD4:C4, C3xC22:Q8, C6xSD16, D4xC2xC6, C3xC22:SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, SD16, C2xD4, C3xD4, C22xC6, C22wrC2, C2xSD16, C8:C22, C3xSD16, C6xD4, C22:SD16, C3xC22wrC2, C6xSD16, C3xC8:C22, C3xC22:SD16
►On 48 points
(1 26 43)(2 27 44)(3 28 45)(4 29 46)(5 30 47)(6 31 48)(7 32 41)(8 25 42)(9 17 36)(10 18 37)(11 19 38)(12 20 39)(13 21 40)(14 22 33)(15 23 34)(16 24 35)
(1 12)(2 6)(3 14)(4 8)(5 16)(7 10)(9 13)(11 15)(17 21)(18 32)(19 23)(20 26)(22 28)(24 30)(25 29)(27 31)(33 45)(34 38)(35 47)(36 40)(37 41)(39 43)(42 46)(44 48)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)(17 29)(18 32)(19 27)(20 30)(21 25)(22 28)(23 31)(24 26)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)
G:=sub<Sym(48)| (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,17,36)(10,18,37)(11,19,38)(12,20,39)(13,21,40)(14,22,33)(15,23,34)(16,24,35), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15)(17,21)(18,32)(19,23)(20,26)(22,28)(24,30)(25,29)(27,31)(33,45)(34,38)(35,47)(36,40)(37,41)(39,43)(42,46)(44,48), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13)(17,29)(18,32)(19,27)(20,30)(21,25)(22,28)(23,31)(24,26)(33,45)(34,48)(35,43)(36,46)(37,41)(38,44)(39,47)(40,42)>;
G:=Group( (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,17,36)(10,18,37)(11,19,38)(12,20,39)(13,21,40)(14,22,33)(15,23,34)(16,24,35), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15)(17,21)(18,32)(19,23)(20,26)(22,28)(24,30)(25,29)(27,31)(33,45)(34,38)(35,47)(36,40)(37,41)(39,43)(42,46)(44,48), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13)(17,29)(18,32)(19,27)(20,30)(21,25)(22,28)(23,31)(24,26)(33,45)(34,48)(35,43)(36,46)(37,41)(38,44)(39,47)(40,42) );
G=PermutationGroup([[(1,26,43),(2,27,44),(3,28,45),(4,29,46),(5,30,47),(6,31,48),(7,32,41),(8,25,42),(9,17,36),(10,18,37),(11,19,38),(12,20,39),(13,21,40),(14,22,33),(15,23,34),(16,24,35)], [(1,12),(2,6),(3,14),(4,8),(5,16),(7,10),(9,13),(11,15),(17,21),(18,32),(19,23),(20,26),(22,28),(24,30),(25,29),(27,31),(33,45),(34,38),(35,47),(36,40),(37,41),(39,43),(42,46),(44,48)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,16),(2,11),(3,14),(4,9),(5,12),(6,15),(7,10),(8,13),(17,29),(18,32),(19,27),(20,30),(21,25),(22,28),(23,31),(24,26),(33,45),(34,48),(35,43),(36,46),(37,41),(38,44),(39,47),(40,42)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | SD16 | C3xD4 | C3xD4 | C3xD4 | C3xSD16 | C8:C22 | C3xC8:C22 |
| kernel | C3xC22:SD16 | C3xC22:C8 | C3xD4:C4 | C3xC22:Q8 | C6xSD16 | D4xC2xC6 | C22:SD16 | C22:C8 | D4:C4 | C22:Q8 | C2xSD16 | C22xD4 | C2xC12 | C3xD4 | C22xC6 | C2xC6 | C2xC4 | D4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 8 | 1 | 2 |
Matrix representation of C3xC22:SD16 ►in GL4(F73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 67 | 6 | 0 | 0 |
| 67 | 67 | 0 | 0 |
| 0 | 0 | 0 | 71 |
| 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[67,67,0,0,6,67,0,0,0,0,0,36,0,0,71,0],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72] >;
C3xC22:SD16 in GAP, Magma, Sage, TeX
C_3\times C_2^2\rtimes {\rm SD}_{16} % in TeX
G:=Group("C3xC2^2:SD16"); // GroupNames label
G:=SmallGroup(192,883);
// by ID
G=gap.SmallGroup(192,883);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations