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, (C2×D4).217D6, C23⋊2D6.5C2, C23.9D6⋊7C2, 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), C23.16D6⋊29C2, C23.23D6⋊18C2, C23.28D6⋊16C2, (C22×S3).35C23, (S3×C23).41C22, C22.125(S3×C23), C23.184(C22×S3), (C22×C6).170C23, (C2×Dic3).207C23, (C22×Dic3).98C22, C4⋊C4⋊S3⋊7C2, (C2×D6⋊C4)⋊22C2, (C4×C3⋊D4)⋊43C2, C2.23(S3×C4○D4), (S3×C22⋊C4)⋊29C2, (C3×C4⋊C4)⋊61C22, 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
Generators and relations for C42⋊18D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 664 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, 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, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.45C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42⋊18D6
►On 48 points
(1 28 10 42)(2 26 11 40)(3 30 12 38)(4 39 7 25)(5 37 8 29)(6 41 9 27)(13 36 22 43)(14 34 23 47)(15 32 24 45)(16 46 19 33)(17 44 20 31)(18 48 21 35)
(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 43 42 33)(26 44 37 34)(27 45 38 35)(28 46 39 36)(29 47 40 31)(30 48 41 32)
(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 30)(26 29)(27 28)(31 47)(32 46)(33 45)(34 44)(35 43)(36 48)(37 40)(38 39)(41 42)
G:=sub<Sym(48)| (1,28,10,42)(2,26,11,40)(3,30,12,38)(4,39,7,25)(5,37,8,29)(6,41,9,27)(13,36,22,43)(14,34,23,47)(15,32,24,45)(16,46,19,33)(17,44,20,31)(18,48,21,35), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,43,42,33)(26,44,37,34)(27,45,38,35)(28,46,39,36)(29,47,40,31)(30,48,41,32), (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)(25,30)(26,29)(27,28)(31,47)(32,46)(33,45)(34,44)(35,43)(36,48)(37,40)(38,39)(41,42)>;
G:=Group( (1,28,10,42)(2,26,11,40)(3,30,12,38)(4,39,7,25)(5,37,8,29)(6,41,9,27)(13,36,22,43)(14,34,23,47)(15,32,24,45)(16,46,19,33)(17,44,20,31)(18,48,21,35), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,43,42,33)(26,44,37,34)(27,45,38,35)(28,46,39,36)(29,47,40,31)(30,48,41,32), (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)(25,30)(26,29)(27,28)(31,47)(32,46)(33,45)(34,44)(35,43)(36,48)(37,40)(38,39)(41,42) );
G=PermutationGroup([[(1,28,10,42),(2,26,11,40),(3,30,12,38),(4,39,7,25),(5,37,8,29),(6,41,9,27),(13,36,22,43),(14,34,23,47),(15,32,24,45),(16,46,19,33),(17,44,20,31),(18,48,21,35)], [(1,16,4,13),(2,17,5,14),(3,18,6,15),(7,22,10,19),(8,23,11,20),(9,24,12,21),(25,43,42,33),(26,44,37,34),(27,45,38,35),(28,46,39,36),(29,47,40,31),(30,48,41,32)], [(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),(25,30),(26,29),(27,28),(31,47),(32,46),(33,45),(34,44),(35,43),(36,48),(37,40),(38,39),(41,42)]])
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 |
Matrix representation of C42⋊18D6 ►in 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] >;
C42⋊18D6 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