metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D12)⋊13C4, (C2×C4).48D12, C42⋊C2⋊5S3, C4.53(D6⋊C4), (C2×Dic6)⋊13C4, (C2×C12).145D4, C22⋊C4.83D6, C22.15(C2×D12), C23.6D6⋊7C2, (C22×C4).130D6, C12.49(C22⋊C4), C23.83(C22×S3), C23.26D6⋊14C2, (C22×C6).112C23, C3⋊2(C23.C23), (C22×C12).155C22, C6.D4.78C22, (S3×C2×C4)⋊3C4, (C2×C4).47(C4×S3), C2.21(C2×D6⋊C4), C22.19(S3×C2×C4), (C2×C12).95(C2×C4), (C2×C4○D12).9C2, (C2×C6).462(C2×D4), C6.48(C2×C22⋊C4), (C3×C42⋊C2)⋊5C2, (C2×C4).46(C3⋊D4), (C22×S3).4(C2×C4), (C2×C6).13(C22×C4), (C2×Dic3).4(C2×C4), C22.28(C2×C3⋊D4), (C2×C3⋊D4).84C22, (C3×C22⋊C4).94C22, SmallGroup(192,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D12)⋊13C4
G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab6, cbc=b-1, bd=db >
Subgroups: 456 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C23⋊C4, C42⋊C2, C42⋊C2, C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C23.C23, C23.6D6, C23.26D6, C3×C42⋊C2, C2×C4○D12, (C2×D12)⋊13C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.C23, C2×D6⋊C4, (C2×D12)⋊13C4
►On 48 points
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 48)(33 47)(34 46)(35 45)(36 44)
(1 19 37 35)(2 20 38 36)(3 21 39 25)(4 22 40 26)(5 23 41 27)(6 24 42 28)(7 13 43 29)(8 14 44 30)(9 15 45 31)(10 16 46 32)(11 17 47 33)(12 18 48 34)
G:=sub<Sym(48)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44), (1,19,37,35)(2,20,38,36)(3,21,39,25)(4,22,40,26)(5,23,41,27)(6,24,42,28)(7,13,43,29)(8,14,44,30)(9,15,45,31)(10,16,46,32)(11,17,47,33)(12,18,48,34)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44), (1,19,37,35)(2,20,38,36)(3,21,39,25)(4,22,40,26)(5,23,41,27)(6,24,42,28)(7,13,43,29)(8,14,44,30)(9,15,45,31)(10,16,46,32)(11,17,47,33)(12,18,48,34) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,48),(33,47),(34,46),(35,45),(36,44)], [(1,19,37,35),(2,20,38,36),(3,21,39,25),(4,22,40,26),(5,23,41,27),(6,24,42,28),(7,13,43,29),(8,14,44,30),(9,15,45,31),(10,16,46,32),(11,17,47,33),(12,18,48,34)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C23.C23 | (C2×D12)⋊13C4 |
| kernel | (C2×D12)⋊13C4 | C23.6D6 | C23.26D6 | C3×C42⋊C2 | C2×C4○D12 | C2×Dic6 | S3×C2×C4 | C2×D12 | C42⋊C2 | C2×C12 | C22⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C3 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 4 | 2 | 1 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of (C2×D12)⋊13C4 ►in GL4(𝔽13) generated by
| 2 | 9 | 8 | 3 |
| 4 | 11 | 5 | 8 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
| 10 | 10 | 0 | 0 |
| 3 | 7 | 0 | 0 |
| 0 | 0 | 7 | 10 |
| 0 | 0 | 3 | 10 |
| 0 | 5 | 6 | 6 |
| 5 | 0 | 6 | 0 |
| 0 | 9 | 8 | 8 |
| 9 | 4 | 0 | 5 |
| 8 | 0 | 4 | 1 |
| 0 | 8 | 3 | 4 |
| 4 | 0 | 5 | 0 |
| 9 | 4 | 0 | 5 |
G:=sub<GL(4,GF(13))| [2,4,0,0,9,11,0,0,8,5,2,9,3,8,4,11],[10,3,0,0,10,7,0,0,0,0,7,3,0,0,10,10],[0,5,0,9,5,0,9,4,6,6,8,0,6,0,8,5],[8,0,4,9,0,8,0,4,4,3,5,0,1,4,0,5] >;
(C2×D12)⋊13C4 in GAP, Magma, Sage, TeX
(C_2\times D_{12})\rtimes_{13}C_4 % in TeX
G:=Group("(C2xD12):13C4"); // GroupNames label
G:=SmallGroup(192,565);
// by ID
G=gap.SmallGroup(192,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1123,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^6,c*b*c=b^-1,b*d=d*b>;
// generators/relations