direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4.10D4, M4(2).20D6, (C4×S3).2D4, C12.95(C2×D4), C4.150(S3×D4), (C2×C12).7C23, (C2×Q8).118D6, (C2×Dic6).4C4, (C6×Q8).5C22, C12.10D4⋊3C2, (S3×M4(2)).1C2, C12.47D4⋊11C2, D6.17(C22⋊C4), C4.Dic3.4C22, Dic3.5(C22⋊C4), (C2×Dic6).46C22, (C3×M4(2)).20C22, (S3×C2×C4).2C4, (C2×S3×Q8).1C2, (C2×C4).6(C4×S3), (C2×C12).6(C2×C4), C3⋊1(C2×C4.10D4), (S3×C2×C4).3C22, C22.16(S3×C2×C4), C6.14(C2×C22⋊C4), C2.15(S3×C22⋊C4), (C2×C4).7(C22×S3), (C3×C4.10D4)⋊9C2, (C2×C6).10(C22×C4), (C2×Dic3).3(C2×C4), (C22×S3).56(C2×C4), SmallGroup(192,309)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4.10D4
G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >
Subgroups: 368 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], S3, C6, C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], Q8 [×8], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×Q8, C2×Q8 [×7], C3⋊C8 [×2], C24 [×2], Dic6 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C4.10D4, C4.10D4 [×3], C2×M4(2) [×2], C22×Q8, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×4], C6×Q8, C2×C4.10D4, C12.47D4 [×2], C12.10D4, C3×C4.10D4, S3×M4(2) [×2], C2×S3×Q8, S3×C4.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C4.10D4 [×2], C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C2×C4.10D4, S3×C22⋊C4, S3×C4.10D4
►On 48 points
(1 28 35)(2 29 36)(3 30 37)(4 31 38)(5 32 39)(6 25 40)(7 26 33)(8 27 34)(9 48 19)(10 41 20)(11 42 21)(12 43 22)(13 44 23)(14 45 24)(15 46 17)(16 47 18)
(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 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 7 10 5 12 3 14)(2 9 4 15 6 13 8 11)(17 40 23 34 21 36 19 38)(18 33 20 39 22 37 24 35)(25 44 27 42 29 48 31 46)(26 41 32 43 30 45 28 47)
G:=sub<Sym(48)| (1,28,35)(2,29,36)(3,30,37)(4,31,38)(5,32,39)(6,25,40)(7,26,33)(8,27,34)(9,48,19)(10,41,20)(11,42,21)(12,43,22)(13,44,23)(14,45,24)(15,46,17)(16,47,18), (17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48), (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), (1,16,7,10,5,12,3,14)(2,9,4,15,6,13,8,11)(17,40,23,34,21,36,19,38)(18,33,20,39,22,37,24,35)(25,44,27,42,29,48,31,46)(26,41,32,43,30,45,28,47)>;
G:=Group( (1,28,35)(2,29,36)(3,30,37)(4,31,38)(5,32,39)(6,25,40)(7,26,33)(8,27,34)(9,48,19)(10,41,20)(11,42,21)(12,43,22)(13,44,23)(14,45,24)(15,46,17)(16,47,18), (17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48), (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), (1,16,7,10,5,12,3,14)(2,9,4,15,6,13,8,11)(17,40,23,34,21,36,19,38)(18,33,20,39,22,37,24,35)(25,44,27,42,29,48,31,46)(26,41,32,43,30,45,28,47) );
G=PermutationGroup([(1,28,35),(2,29,36),(3,30,37),(4,31,38),(5,32,39),(6,25,40),(7,26,33),(8,27,34),(9,48,19),(10,41,20),(11,42,21),(12,43,22),(13,44,23),(14,45,24),(15,46,17),(16,47,18)], [(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12),(17,19,21,23),(18,24,22,20),(25,31,29,27),(26,28,30,32),(33,35,37,39),(34,40,38,36),(41,47,45,43),(42,44,46,48)], [(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)], [(1,16,7,10,5,12,3,14),(2,9,4,15,6,13,8,11),(17,40,23,34,21,36,19,38),(18,33,20,39,22,37,24,35),(25,44,27,42,29,48,31,46),(26,41,32,43,30,45,28,47)])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | C4.10D4 | S3×D4 | S3×C4.10D4 |
| kernel | S3×C4.10D4 | C12.47D4 | C12.10D4 | C3×C4.10D4 | S3×M4(2) | C2×S3×Q8 | C2×Dic6 | S3×C2×C4 | C4.10D4 | C4×S3 | M4(2) | C2×Q8 | C2×C4 | S3 | C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 |
Matrix representation of S3×C4.10D4 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 16 |
| 0 | 0 | 0 | 0 | 16 | 20 |
| 0 | 0 | 57 | 53 | 0 | 0 |
| 0 | 0 | 53 | 16 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,57,53,0,0,0,0,53,16,0,0,53,16,0,0,0,0,16,20,0,0] >;
S3×C4.10D4 in GAP, Magma, Sage, TeX
S_3\times C_4._{10}D_4 % in TeX
G:=Group("S3xC4.10D4"); // GroupNames label
G:=SmallGroup(192,309);
// by ID
G=gap.SmallGroup(192,309);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,58,570,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^4=c^2,e^2=d*c*d^-1=c^-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,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations