metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D4)⋊43D6, (C2×C12)⋊15D4, (C2×Q8)⋊35D6, D6⋊6(C4○D4), (C22×C4)⋊32D6, C23⋊2D6⋊34C2, D6⋊3D4⋊46C2, D6⋊3Q8⋊47C2, C12.265(C2×D4), (C6×D4)⋊47C22, (C6×Q8)⋊39C22, (C2×C6).314C24, C4⋊Dic3⋊80C22, C6.164(C22×D4), (C2×C12).651C23, Dic3⋊C4⋊76C22, D6⋊C4.159C22, (C22×C12)⋊42C22, C3⋊8(C22.19C24), (C4×Dic3)⋊60C22, C23.26D6⋊38C2, C23.23D6⋊34C2, C6.D4⋊65C22, (C22×C6).240C23, C22.323(S3×C23), C23.147(C22×S3), (C22×S3).243C23, (S3×C23).115C22, (C2×Dic3).293C23, (C22×Dic3).236C22, (C6×C4○D4)⋊6C2, (S3×C22×C4)⋊7C2, (C2×C4○D4)⋊10S3, (C4×C3⋊D4)⋊60C2, (C2×C6).80(C2×D4), (C2×C4)⋊14(C3⋊D4), C2.103(S3×C4○D4), C6.215(C2×C4○D4), C4.100(C2×C3⋊D4), (S3×C2×C4).263C22, C2.37(C22×C3⋊D4), C22.23(C2×C3⋊D4), (C2×C4).639(C22×S3), (C2×C3⋊D4).140C22, SmallGroup(192,1387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D4)⋊43D6
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b2c, ede=d-1 >
Subgroups: 808 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C22.19C24, C23.26D6, C4×C3⋊D4, C23.23D6, C23⋊2D6, D6⋊3D4, D6⋊3Q8, S3×C22×C4, C6×C4○D4, (C2×D4)⋊43D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, C2×C3⋊D4, S3×C23, C22.19C24, S3×C4○D4, C22×C3⋊D4, (C2×D4)⋊43D6
►On 48 points
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 20)(14 21)(15 19)(16 24)(17 22)(18 23)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
(1 19 4 23)(2 20 5 24)(3 21 6 22)(7 18 10 15)(8 16 11 13)(9 17 12 14)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 43 34 46)(32 44 35 47)(33 45 36 48)
(1 40)(2 38)(3 42)(4 37)(5 41)(6 39)(7 33)(8 31)(9 35)(10 36)(11 34)(12 32)(13 43)(14 47)(15 45)(16 46)(17 44)(18 48)(19 25)(20 29)(21 27)(22 30)(23 28)(24 26)
(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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 16)(14 18)(15 17)(19 22)(20 24)(21 23)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)
G:=sub<Sym(48)| (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,20)(14,21)(15,19)(16,24)(17,22)(18,23)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (1,19,4,23)(2,20,5,24)(3,21,6,22)(7,18,10,15)(8,16,11,13)(9,17,12,14)(25,40,28,37)(26,41,29,38)(27,42,30,39)(31,43,34,46)(32,44,35,47)(33,45,36,48), (1,40)(2,38)(3,42)(4,37)(5,41)(6,39)(7,33)(8,31)(9,35)(10,36)(11,34)(12,32)(13,43)(14,47)(15,45)(16,46)(17,44)(18,48)(19,25)(20,29)(21,27)(22,30)(23,28)(24,26), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,16)(14,18)(15,17)(19,22)(20,24)(21,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39)>;
G:=Group( (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,20)(14,21)(15,19)(16,24)(17,22)(18,23)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (1,19,4,23)(2,20,5,24)(3,21,6,22)(7,18,10,15)(8,16,11,13)(9,17,12,14)(25,40,28,37)(26,41,29,38)(27,42,30,39)(31,43,34,46)(32,44,35,47)(33,45,36,48), (1,40)(2,38)(3,42)(4,37)(5,41)(6,39)(7,33)(8,31)(9,35)(10,36)(11,34)(12,32)(13,43)(14,47)(15,45)(16,46)(17,44)(18,48)(19,25)(20,29)(21,27)(22,30)(23,28)(24,26), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,16)(14,18)(15,17)(19,22)(20,24)(21,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,20),(14,21),(15,19),(16,24),(17,22),(18,23),(25,45),(26,46),(27,47),(28,48),(29,43),(30,44),(31,41),(32,42),(33,37),(34,38),(35,39),(36,40)], [(1,19,4,23),(2,20,5,24),(3,21,6,22),(7,18,10,15),(8,16,11,13),(9,17,12,14),(25,40,28,37),(26,41,29,38),(27,42,30,39),(31,43,34,46),(32,44,35,47),(33,45,36,48)], [(1,40),(2,38),(3,42),(4,37),(5,41),(6,39),(7,33),(8,31),(9,35),(10,36),(11,34),(12,32),(13,43),(14,47),(15,45),(16,46),(17,44),(18,48),(19,25),(20,29),(21,27),(22,30),(23,28),(24,26)], [(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,16),(14,18),(15,17),(19,22),(20,24),(21,23),(25,44),(26,43),(27,48),(28,47),(29,46),(30,45),(31,38),(32,37),(33,42),(34,41),(35,40),(36,39)]])
48 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 | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | S3×C4○D4 |
| kernel | (C2×D4)⋊43D6 | C23.26D6 | C4×C3⋊D4 | C23.23D6 | C23⋊2D6 | D6⋊3D4 | D6⋊3Q8 | S3×C22×C4 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C4 | C2×D4 | C2×Q8 | D6 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 8 | 8 | 4 |
Matrix representation of (C2×D4)⋊43D6 ►in 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,10,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12] >;
(C2×D4)⋊43D6 in GAP, Magma, Sage, TeX
(C_2\times D_4)\rtimes_{43}D_6 % in TeX
G:=Group("(C2xD4):43D6"); // GroupNames label
G:=SmallGroup(192,1387);
// by ID
G=gap.SmallGroup(192,1387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*d*e=d^-1>;
// generators/relations