metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.2C4≀C2, C4⋊Dic3⋊2C4, (C2×Dic6)⋊1C4, (C2×C6).13Q16, (C2×C4).104D12, (C2×C12).222D4, C6.2(C23⋊C4), (C2×C6).28SD16, (C22×C4).69D6, C6.1(Q8⋊C4), (C22×C6).175D4, C2.5(C42⋊4S3), C22.55(D6⋊C4), C12.55D4.1C2, C12.48D4.7C2, C2.3(C6.SD16), C23.78(C3⋊D4), C3⋊1(C23.31D4), C2.C42.7S3, C22.4(D4.S3), C22.4(C3⋊Q16), (C22×C12).90C22, C2.5(C23.6D6), (C2×C4).9(C4×S3), (C2×C12).21(C2×C4), (C2×C6).34(C22⋊C4), (C3×C2.C42).13C2, SmallGroup(192,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic3⋊C4
G = < a,b,c,d | a4=b6=d4=1, c2=b3, ab=ba, cac-1=a-1, dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=a-1b3c >
Subgroups: 224 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C22⋊C8, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C22×C12, C23.31D4, C12.55D4, C3×C2.C42, C12.48D4, C4⋊Dic3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C3⋊D4, C23⋊C4, Q8⋊C4, C4≀C2, D6⋊C4, D4.S3, C3⋊Q16, C23.31D4, C42⋊4S3, C23.6D6, C6.SD16, C4⋊Dic3⋊C4
►On 48 points
(1 22 10 13)(2 23 11 14)(3 24 12 15)(4 19 7 16)(5 20 8 17)(6 21 9 18)(25 37 34 46)(26 38 35 47)(27 39 36 48)(28 40 31 43)(29 41 32 44)(30 42 33 45)
(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 26 4 29)(2 25 5 28)(3 30 6 27)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
G:=sub<Sym(48)| (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (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), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,32,10,35)(8,31,11,34)(9,36,12,33)(13,38,16,41)(14,37,17,40)(15,42,18,39)(19,44,22,47)(20,43,23,46)(21,48,24,45), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
G:=Group( (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (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), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,32,10,35)(8,31,11,34)(9,36,12,33)(13,38,16,41)(14,37,17,40)(15,42,18,39)(19,44,22,47)(20,43,23,46)(21,48,24,45), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
G=PermutationGroup([[(1,22,10,13),(2,23,11,14),(3,24,12,15),(4,19,7,16),(5,20,8,17),(6,21,9,18),(25,37,34,46),(26,38,35,47),(27,39,36,48),(28,40,31,43),(29,41,32,44),(30,42,33,45)], [(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)], [(1,26,4,29),(2,25,5,28),(3,30,6,27),(7,32,10,35),(8,31,11,34),(9,36,12,33),(13,38,16,41),(14,37,17,40),(15,42,18,39),(19,44,22,47),(20,43,23,46),(21,48,24,45)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C42⋊4S3 | C23⋊C4 | D4.S3 | C3⋊Q16 | C23.6D6 |
| kernel | C4⋊Dic3⋊C4 | C12.55D4 | C3×C2.C42 | C12.48D4 | C4⋊Dic3 | C2×Dic6 | C2.C42 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C6 | C2 | C6 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of C4⋊Dic3⋊C4 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 64 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 63 | 46 |
| 8 | 0 | 0 | 0 |
| 15 | 64 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 12 | 1 | 0 | 0 |
| 3 | 61 | 0 | 0 |
| 0 | 0 | 22 | 2 |
| 0 | 0 | 13 | 51 |
| 1 | 0 | 0 | 0 |
| 29 | 46 | 0 | 0 |
| 0 | 0 | 1 | 20 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [46,64,0,0,0,27,0,0,0,0,27,63,0,0,0,46],[8,15,0,0,0,64,0,0,0,0,72,0,0,0,0,72],[12,3,0,0,1,61,0,0,0,0,22,13,0,0,2,51],[1,29,0,0,0,46,0,0,0,0,1,0,0,0,20,72] >;
C4⋊Dic3⋊C4 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C4:Dic3:C4"); // GroupNames label
G:=SmallGroup(192,11);
// by ID
G=gap.SmallGroup(192,11);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,1571,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^6=d^4=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^-1*b^3*c>;
// generators/relations