direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C22⋊Q8, C4⋊C4⋊25D6, D6⋊3(C2×Q8), (C2×Q8)⋊17D6, C22⋊3(S3×Q8), D6.59(C2×D4), (C4×S3).41D4, C4.185(S3×D4), (C6×Q8)⋊5C22, (C22×S3)⋊6Q8, D6⋊Q8⋊17C2, C4.D12⋊23C2, D6⋊3Q8⋊12C2, C12.230(C2×D4), C22⋊C4.55D6, C6.72(C22×D4), D6.38(C4○D4), C6.34(C22×Q8), (C2×C12).51C23, (C2×C6).170C24, D6⋊C4.20C22, C4⋊Dic3⋊34C22, Dic3.47(C2×D4), (C22×C4).387D6, C12.48D4⋊35C2, Dic3⋊C4⋊30C22, (C2×Dic6)⋊27C22, Dic3.D4⋊22C2, (C22×C6).198C23, C23.197(C22×S3), C22.191(S3×C23), (C2×Dic3).85C23, (S3×C23).108C22, (C22×S3).192C23, (C22×C12).250C22, C6.D4.32C22, (C22×Dic3).224C22, (C2×S3×Q8)⋊5C2, (S3×C4⋊C4)⋊24C2, (C2×C6)⋊2(C2×Q8), C2.45(C2×S3×D4), C3⋊4(C2×C22⋊Q8), C2.17(C2×S3×Q8), C2.47(S3×C4○D4), (S3×C22×C4).7C2, (C3×C22⋊Q8)⋊6C2, (C3×C4⋊C4)⋊17C22, C6.159(C2×C4○D4), (S3×C22⋊C4).1C2, (S3×C2×C4).92C22, (C2×C4).45(C22×S3), (C3×C22⋊C4).25C22, SmallGroup(192,1185)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 800 in 322 conjugacy classes, 121 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×20], S3 [×4], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×28], Q8 [×8], C23, C23 [×10], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×5], D6 [×8], D6 [×10], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×13], C2×Q8, C2×Q8 [×7], C24, Dic6 [×6], C4×S3 [×4], C4×S3 [×14], C2×Dic3 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C22×S3 [×4], C22×S3 [×4], C22×C6, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8, C22⋊Q8 [×7], C23×C4, C22×Q8, Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], S3×Q8 [×4], C22×Dic3, C22×C12, C6×Q8, S3×C23, C2×C22⋊Q8, Dic3.D4 [×2], S3×C22⋊C4 [×2], S3×C4⋊C4, S3×C4⋊C4 [×2], D6⋊Q8 [×2], C4.D12, C12.48D4, D6⋊3Q8, C3×C22⋊Q8, S3×C22×C4, C2×S3×Q8, S3×C22⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, S3×D4 [×2], S3×Q8 [×2], S3×C23, C2×C22⋊Q8, C2×S3×D4, C2×S3×Q8, S3×C4○D4, S3×C22⋊Q8
Generators and relations
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >
►On 48 points
(1 32 47)(2 29 48)(3 30 45)(4 31 46)(5 44 34)(6 41 35)(7 42 36)(8 43 33)(9 19 21)(10 20 22)(11 17 23)(12 18 24)(13 25 37)(14 26 38)(15 27 39)(16 28 40)
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)(17 45)(18 46)(19 47)(20 48)(21 32)(22 29)(23 30)(24 31)(25 33)(26 34)(27 35)(28 36)(37 43)(38 44)(39 41)(40 42)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 32)(18 29)(19 30)(20 31)(21 45)(22 46)(23 47)(24 48)(25 27)(26 28)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)(17 30)(18 31)(19 32)(20 29)(21 47)(22 48)(23 45)(24 46)(25 43)(26 44)(27 41)(28 42)(33 37)(34 38)(35 39)(36 40)
(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 5 3 7)(2 8 4 6)(9 14 11 16)(10 13 12 15)(17 28 19 26)(18 27 20 25)(21 38 23 40)(22 37 24 39)(29 43 31 41)(30 42 32 44)(33 46 35 48)(34 45 36 47)
G:=sub<Sym(48)| (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,34)(6,41,35)(7,42,36)(8,43,33)(9,19,21)(10,20,22)(11,17,23)(12,18,24)(13,25,37)(14,26,38)(15,27,39)(16,28,40), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,45)(18,46)(19,47)(20,48)(21,32)(22,29)(23,30)(24,31)(25,33)(26,34)(27,35)(28,36)(37,43)(38,44)(39,41)(40,42), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,45)(22,46)(23,47)(24,48)(25,27)(26,28)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,30)(18,31)(19,32)(20,29)(21,47)(22,48)(23,45)(24,46)(25,43)(26,44)(27,41)(28,42)(33,37)(34,38)(35,39)(36,40), (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), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15)(17,28,19,26)(18,27,20,25)(21,38,23,40)(22,37,24,39)(29,43,31,41)(30,42,32,44)(33,46,35,48)(34,45,36,47)>;
G:=Group( (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,34)(6,41,35)(7,42,36)(8,43,33)(9,19,21)(10,20,22)(11,17,23)(12,18,24)(13,25,37)(14,26,38)(15,27,39)(16,28,40), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,45)(18,46)(19,47)(20,48)(21,32)(22,29)(23,30)(24,31)(25,33)(26,34)(27,35)(28,36)(37,43)(38,44)(39,41)(40,42), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,45)(22,46)(23,47)(24,48)(25,27)(26,28)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,30)(18,31)(19,32)(20,29)(21,47)(22,48)(23,45)(24,46)(25,43)(26,44)(27,41)(28,42)(33,37)(34,38)(35,39)(36,40), (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), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15)(17,28,19,26)(18,27,20,25)(21,38,23,40)(22,37,24,39)(29,43,31,41)(30,42,32,44)(33,46,35,48)(34,45,36,47) );
G=PermutationGroup([(1,32,47),(2,29,48),(3,30,45),(4,31,46),(5,44,34),(6,41,35),(7,42,36),(8,43,33),(9,19,21),(10,20,22),(11,17,23),(12,18,24),(13,25,37),(14,26,38),(15,27,39),(16,28,40)], [(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13),(17,45),(18,46),(19,47),(20,48),(21,32),(22,29),(23,30),(24,31),(25,33),(26,34),(27,35),(28,36),(37,43),(38,44),(39,41),(40,42)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,32),(18,29),(19,30),(20,31),(21,45),(22,46),(23,47),(24,48),(25,27),(26,28),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44)], [(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13),(17,30),(18,31),(19,32),(20,29),(21,47),(22,48),(23,45),(24,46),(25,43),(26,44),(27,41),(28,42),(33,37),(34,38),(35,39),(36,40)], [(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)], [(1,5,3,7),(2,8,4,6),(9,14,11,16),(10,13,12,15),(17,28,19,26),(18,27,20,25),(21,38,23,40),(22,37,24,39),(29,43,31,41),(30,42,32,44),(33,46,35,48),(34,45,36,47)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 12 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 5 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 4 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,3,0,0,0,0,5,6,0,0,0,0,0,0,5,10,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,4,0,0,0,0,6,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
42 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 | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| 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 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | D6 | C4○D4 | S3×D4 | S3×Q8 | S3×C4○D4 |
| kernel | S3×C22⋊Q8 | Dic3.D4 | S3×C22⋊C4 | S3×C4⋊C4 | D6⋊Q8 | C4.D12 | C12.48D4 | D6⋊3Q8 | C3×C22⋊Q8 | S3×C22×C4 | C2×S3×Q8 | C22⋊Q8 | C4×S3 | C22×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | D6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 3 | 1 | 1 | 4 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
S_3\times C_2^2\rtimes Q_8
% in TeX
G:=Group("S3xC2^2:Q8"); // GroupNames label
G:=SmallGroup(192,1185);
// by ID
G=gap.SmallGroup(192,1185);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,794,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=e^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations