metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊18D6, C6.182+ (1+4), C4⋊C4⋊49D6, (C4×D4)⋊18S3, (C22×C4)⋊8D6, (D4×C12)⋊20C2, D6⋊Q8⋊8C2, C22⋊C4⋊48D6, (C4×C12)⋊32C22, C23⋊2D6.5C2, C23.9D6⋊7C2, (C2×D4).217D6, C4⋊Dic3⋊9C22, C42⋊3S3⋊16C2, C42⋊2S3⋊32C2, D6.17(C4○D4), (C2×C6).100C24, D6⋊C4.85C22, (C2×Dic6)⋊6C22, C12.48D4⋊11C2, C2.19(D4⋊6D6), (C2×C12).699C23, Dic3⋊C4⋊42C22, (C22×C12)⋊37C22, C23.11D6⋊7C2, (C4×Dic3)⋊52C22, (C6×D4).307C22, C6.D4⋊9C22, C3⋊3(C22.45C24), C22.12(C4○D12), C23.28D6⋊16C2, C23.16D6⋊29C2, C23.23D6⋊18C2, (S3×C23).41C22, (C22×S3).35C23, C23.184(C22×S3), (C22×C6).170C23, C22.125(S3×C23), (C2×Dic3).207C23, (C22×Dic3).98C22, C4⋊C4⋊S3⋊7C2, (C2×D6⋊C4)⋊22C2, (C4×C3⋊D4)⋊43C2, C2.23(S3×C4○D4), (C3×C4⋊C4)⋊61C22, (S3×C22⋊C4)⋊29C2, C2.49(C2×C4○D12), C6.140(C2×C4○D4), (C2×C6).16(C4○D4), (S3×C2×C4).201C22, (C3×C22⋊C4)⋊57C22, (C2×C4).284(C22×S3), (C2×C3⋊D4).16C22, SmallGroup(192,1115)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 664 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×2], C22 [×16], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×2], D6 [×9], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×8], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C22×Dic3, C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23, C22.45C24, C42⋊2S3, C42⋊3S3, C23.16D6, S3×C22⋊C4, C23.9D6, C23.11D6, D6⋊Q8, C4⋊C4⋊S3, C12.48D4, C2×D6⋊C4, C4×C3⋊D4, C23.28D6, C23.23D6, C23⋊2D6, D4×C12, C42⋊18D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), C4○D12 [×2], S3×C23, C22.45C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42⋊18D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
►On 48 points
(1 34 10 40)(2 32 11 38)(3 36 12 42)(4 37 7 31)(5 41 8 35)(6 39 9 33)(13 25 22 43)(14 29 23 47)(15 27 24 45)(16 46 19 28)(17 44 20 26)(18 48 21 30)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 34 46 37)(26 35 47 38)(27 36 48 39)(28 31 43 40)(29 32 44 41)(30 33 45 42)
(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 3)(4 6)(7 9)(10 12)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)
G:=sub<Sym(48)| (1,34,10,40)(2,32,11,38)(3,36,12,42)(4,37,7,31)(5,41,8,35)(6,39,9,33)(13,25,22,43)(14,29,23,47)(15,27,24,45)(16,46,19,28)(17,44,20,26)(18,48,21,30), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,34,46,37)(26,35,47,38)(27,36,48,39)(28,31,43,40)(29,32,44,41)(30,33,45,42), (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), (1,3)(4,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)>;
G:=Group( (1,34,10,40)(2,32,11,38)(3,36,12,42)(4,37,7,31)(5,41,8,35)(6,39,9,33)(13,25,22,43)(14,29,23,47)(15,27,24,45)(16,46,19,28)(17,44,20,26)(18,48,21,30), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,34,46,37)(26,35,47,38)(27,36,48,39)(28,31,43,40)(29,32,44,41)(30,33,45,42), (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), (1,3)(4,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40) );
G=PermutationGroup([(1,34,10,40),(2,32,11,38),(3,36,12,42),(4,37,7,31),(5,41,8,35),(6,39,9,33),(13,25,22,43),(14,29,23,47),(15,27,24,45),(16,46,19,28),(17,44,20,26),(18,48,21,30)], [(1,16,4,13),(2,17,5,14),(3,18,6,15),(7,22,10,19),(8,23,11,20),(9,24,12,21),(25,34,46,37),(26,35,47,38),(27,36,48,39),(28,31,43,40),(29,32,44,41),(30,33,45,42)], [(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)], [(1,3),(4,6),(7,9),(10,12),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,36),(32,35),(33,34),(37,42),(38,41),(39,40)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 12 | 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 | 0 |
| 0 | 0 | 0 | 0 | 8 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | C42⋊18D6 | C42⋊2S3 | C42⋊3S3 | C23.16D6 | S3×C22⋊C4 | C23.9D6 | C23.11D6 | D6⋊Q8 | C4⋊C4⋊S3 | C12.48D4 | C2×D6⋊C4 | C4×C3⋊D4 | C23.28D6 | C23.23D6 | C23⋊2D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C2×C6 | C22 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}D_6 % in TeX
G:=Group("C4^2:18D6"); // GroupNames label
G:=SmallGroup(192,1115);
// by ID
G=gap.SmallGroup(192,1115);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations