direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×M4(2)⋊S3, M4(2)⋊22D6, C4.65(C2×D12), (C2×C4).49D12, C4.28(D6⋊C4), (C2×D12).15C4, C12.416(C2×D4), (C2×C12).172D4, C6⋊1(C4.D4), (S3×C23).3C4, C23.60(C4×S3), (C2×M4(2))⋊10S3, (C6×M4(2))⋊18C2, (C22×C4).154D6, C12.53(C22⋊C4), (C2×C12).416C23, C22.50(D6⋊C4), (C22×D12).14C2, C4.Dic3⋊21C22, (C2×D12).250C22, (C3×M4(2))⋊34C22, (C22×C12).187C22, C3⋊2(C2×C4.D4), (C2×C4).52(C4×S3), C2.29(C2×D6⋊C4), C22.20(S3×C2×C4), C4.109(C2×C3⋊D4), C6.57(C2×C22⋊C4), (C2×C12).107(C2×C4), (C22×S3).5(C2×C4), (C2×C4.Dic3)⋊15C2, (C22×C6).70(C2×C4), (C2×C6).14(C22×C4), (C2×C4).256(C3⋊D4), (C2×C6).65(C22⋊C4), (C2×C4).120(C22×S3), SmallGroup(192,689)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×M4(2)⋊S3
G = < a,b,c,d,e | a2=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >
Subgroups: 664 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C22 [×3], C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×6], D4 [×8], C23, C23 [×12], C12 [×4], D6 [×16], C2×C6 [×3], C2×C6 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C2×D4 [×8], C24 [×2], C3⋊C8 [×2], C24 [×2], D12 [×8], C2×C12 [×6], C22×S3 [×4], C22×S3 [×8], C22×C6, C4.D4 [×4], C2×M4(2), C2×M4(2), C22×D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2), C2×D12 [×4], C2×D12 [×4], C22×C12, S3×C23 [×2], C2×C4.D4, M4(2)⋊S3 [×4], C2×C4.Dic3, C6×M4(2), C22×D12, C2×M4(2)⋊S3
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], D12 [×2], C3⋊D4 [×2], C22×S3, C4.D4 [×2], C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4.D4, M4(2)⋊S3 [×2], C2×D6⋊C4, C2×M4(2)⋊S3
►On 48 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(33 44)(34 41)(35 46)(36 43)(37 48)(38 45)(39 42)(40 47)
(1 47 11)(2 48 12)(3 41 13)(4 42 14)(5 43 15)(6 44 16)(7 45 9)(8 46 10)(17 38 32)(18 39 25)(19 40 26)(20 33 27)(21 34 28)(22 35 29)(23 36 30)(24 37 31)
(1 21)(2 8)(3 19)(4 6)(5 17)(7 23)(9 36)(10 48)(11 34)(12 46)(13 40)(14 44)(15 38)(16 42)(18 20)(22 24)(25 33)(26 41)(27 39)(28 47)(29 37)(30 45)(31 35)(32 43)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(33,44)(34,41)(35,46)(36,43)(37,48)(38,45)(39,42)(40,47), (1,47,11)(2,48,12)(3,41,13)(4,42,14)(5,43,15)(6,44,16)(7,45,9)(8,46,10)(17,38,32)(18,39,25)(19,40,26)(20,33,27)(21,34,28)(22,35,29)(23,36,30)(24,37,31), (1,21)(2,8)(3,19)(4,6)(5,17)(7,23)(9,36)(10,48)(11,34)(12,46)(13,40)(14,44)(15,38)(16,42)(18,20)(22,24)(25,33)(26,41)(27,39)(28,47)(29,37)(30,45)(31,35)(32,43)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(33,44)(34,41)(35,46)(36,43)(37,48)(38,45)(39,42)(40,47), (1,47,11)(2,48,12)(3,41,13)(4,42,14)(5,43,15)(6,44,16)(7,45,9)(8,46,10)(17,38,32)(18,39,25)(19,40,26)(20,33,27)(21,34,28)(22,35,29)(23,36,30)(24,37,31), (1,21)(2,8)(3,19)(4,6)(5,17)(7,23)(9,36)(10,48)(11,34)(12,46)(13,40)(14,44)(15,38)(16,42)(18,20)(22,24)(25,33)(26,41)(27,39)(28,47)(29,37)(30,45)(31,35)(32,43) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(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)], [(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,32),(10,29),(11,26),(12,31),(13,28),(14,25),(15,30),(16,27),(33,44),(34,41),(35,46),(36,43),(37,48),(38,45),(39,42),(40,47)], [(1,47,11),(2,48,12),(3,41,13),(4,42,14),(5,43,15),(6,44,16),(7,45,9),(8,46,10),(17,38,32),(18,39,25),(19,40,26),(20,33,27),(21,34,28),(22,35,29),(23,36,30),(24,37,31)], [(1,21),(2,8),(3,19),(4,6),(5,17),(7,23),(9,36),(10,48),(11,34),(12,46),(13,40),(14,44),(15,38),(16,42),(18,20),(22,24),(25,33),(26,41),(27,39),(28,47),(29,37),(30,45),(31,35),(32,43)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | C4.D4 | M4(2)⋊S3 |
| kernel | C2×M4(2)⋊S3 | M4(2)⋊S3 | C2×C4.Dic3 | C6×M4(2) | C22×D12 | C2×D12 | S3×C23 | C2×M4(2) | C2×C12 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C2×M4(2)⋊S3 ►in GL6(𝔽73)
| 72 | 0 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 66 | 14 | 0 | 0 |
| 0 | 0 | 59 | 7 | 0 | 0 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 7 | 0 | 0 |
| 0 | 0 | 66 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 0 | 0 | 66 | 14 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,59,66,0,0,0,0,7,14,0,0,0,0,0,0,59,66,0,0,0,0,7,14] >;
C2×M4(2)⋊S3 in GAP, Magma, Sage, TeX
C_2\times M_{4(2})\rtimes S_3 % in TeX
G:=Group("C2xM4(2):S3"); // GroupNames label
G:=SmallGroup(192,689);
// by ID
G=gap.SmallGroup(192,689);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations