direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C22⋊C8, D6.4M4(2), D6⋊5(C2×C8), (C2×C8)⋊24D6, D6⋊C8⋊20C2, C22⋊4(S3×C8), (C22×S3)⋊2C8, (C4×S3).52D4, C4.193(S3×D4), C6.6(C22×C8), (C2×C24)⋊25C22, C12.352(C2×D4), (S3×C23).4C4, C23.51(C4×S3), C2.4(S3×M4(2)), (C22×C4).318D6, C6.21(C2×M4(2)), C12.55D4⋊22C2, (C2×C12).819C23, D6.14(C22⋊C4), (C22×Dic3).9C4, Dic3.15(C22⋊C4), (C22×C12).336C22, C2.8(S3×C2×C8), (S3×C2×C8)⋊12C2, (C2×C6)⋊1(C2×C8), C3⋊1(C2×C22⋊C8), (S3×C2×C4).17C4, (C2×C3⋊C8)⋊43C22, C2.3(S3×C22⋊C4), C6.7(C2×C22⋊C4), C22.43(S3×C2×C4), (C3×C22⋊C8)⋊18C2, (C2×C4).131(C4×S3), (S3×C22×C4).16C2, (C2×C12).152(C2×C4), (S3×C2×C4).306C22, (C2×C6).74(C22×C4), (C22×C6).37(C2×C4), (C22×S3).54(C2×C4), (C2×C4).761(C22×S3), (C2×Dic3).84(C2×C4), SmallGroup(192,283)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C22⋊C8
G = < a,b,c,d,e | a3=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 512 in 202 conjugacy classes, 73 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C22×C8, C23×C4, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C22⋊C8, D6⋊C8, C12.55D4, C3×C22⋊C8, S3×C2×C8, S3×C22×C4, S3×C22⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C4×S3, C22×S3, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), S3×C8, S3×C2×C4, S3×D4, C2×C22⋊C8, S3×C22⋊C4, S3×C2×C8, S3×M4(2), S3×C22⋊C8
►On 48 points
(1 19 43)(2 20 44)(3 21 45)(4 22 46)(5 23 47)(6 24 48)(7 17 41)(8 18 42)(9 30 37)(10 31 38)(11 32 39)(12 25 40)(13 26 33)(14 27 34)(15 28 35)(16 29 36)
(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 5)(2 16)(3 7)(4 10)(6 12)(8 14)(9 13)(11 15)(17 21)(18 27)(19 23)(20 29)(22 31)(24 25)(26 30)(28 32)(33 37)(34 42)(35 39)(36 44)(38 46)(40 48)(41 45)(43 47)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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)
G:=sub<Sym(48)| (1,19,43)(2,20,44)(3,21,45)(4,22,46)(5,23,47)(6,24,48)(7,17,41)(8,18,42)(9,30,37)(10,31,38)(11,32,39)(12,25,40)(13,26,33)(14,27,34)(15,28,35)(16,29,36), (17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15)(17,21)(18,27)(19,23)(20,29)(22,31)(24,25)(26,30)(28,32)(33,37)(34,42)(35,39)(36,44)(38,46)(40,48)(41,45)(43,47), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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)>;
G:=Group( (1,19,43)(2,20,44)(3,21,45)(4,22,46)(5,23,47)(6,24,48)(7,17,41)(8,18,42)(9,30,37)(10,31,38)(11,32,39)(12,25,40)(13,26,33)(14,27,34)(15,28,35)(16,29,36), (17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15)(17,21)(18,27)(19,23)(20,29)(22,31)(24,25)(26,30)(28,32)(33,37)(34,42)(35,39)(36,44)(38,46)(40,48)(41,45)(43,47), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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) );
G=PermutationGroup([[(1,19,43),(2,20,44),(3,21,45),(4,22,46),(5,23,47),(6,24,48),(7,17,41),(8,18,42),(9,30,37),(10,31,38),(11,32,39),(12,25,40),(13,26,33),(14,27,34),(15,28,35),(16,29,36)], [(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(1,5),(2,16),(3,7),(4,10),(6,12),(8,14),(9,13),(11,15),(17,21),(18,27),(19,23),(20,29),(22,31),(24,25),(26,30),(28,32),(33,37),(34,42),(35,39),(36,44),(38,46),(40,48),(41,45),(43,47)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 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 | C4 | C4 | C4 | C8 | S3 | D4 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | S3×C8 | S3×D4 | S3×M4(2) |
| kernel | S3×C22⋊C8 | D6⋊C8 | C12.55D4 | C3×C22⋊C8 | S3×C2×C8 | S3×C22×C4 | S3×C2×C4 | C22×Dic3 | S3×C23 | C22×S3 | C22⋊C8 | C4×S3 | C2×C8 | C22×C4 | D6 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 16 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of S3×C22⋊C8 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 27 | 0 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[22,0,0,0,0,22,0,0,0,0,0,27,0,0,1,0] >;
S3×C22⋊C8 in GAP, Magma, Sage, TeX
S_3\times C_2^2\rtimes C_8
% in TeX
G:=Group("S3xC2^2:C8"); // GroupNames label
G:=SmallGroup(192,283);
// by ID
G=gap.SmallGroup(192,283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations