metabelian, supersoluble, monomial
Aliases: C62⋊13D4, C62.255C23, (C6×D4)⋊14S3, (C2×C12)⋊16D6, (C22×C6)⋊7D6, C6.129(S3×D4), C3⋊4(C23⋊2D6), (C6×C12)⋊25C22, C32⋊11C22≀C2, (C2×C62)⋊5C22, C62⋊5C4⋊18C2, C6.11D12⋊25C2, C22⋊3(C32⋊7D4), (D4×C3×C6)⋊17C2, (C2×C3⋊S3)⋊17D4, C23⋊2(C2×C3⋊S3), C2.25(D4×C3⋊S3), (C2×D4)⋊3(C3⋊S3), (C2×C6)⋊8(C3⋊D4), (C23×C3⋊S3)⋊3C2, (C3×C6).283(C2×D4), C6.124(C2×C3⋊D4), (C2×C32⋊7D4)⋊10C2, (C2×C3⋊Dic3)⋊8C22, C2.13(C2×C32⋊7D4), (C2×C6).272(C22×S3), C22.59(C22×C3⋊S3), (C22×C3⋊S3).92C22, (C2×C4)⋊2(C2×C3⋊S3), SmallGroup(288,794)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊13D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1772 in 390 conjugacy classes, 81 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C2×D4, C24, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22≀C2, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C2×C3⋊Dic3, C32⋊7D4, C6×C12, D4×C32, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C23⋊2D6, C6.11D12, C62⋊5C4, C2×C32⋊7D4, D4×C3×C6, C23×C3⋊S3, C62⋊13D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C22≀C2, C2×C3⋊S3, S3×D4, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C23⋊2D6, D4×C3⋊S3, C2×C32⋊7D4, C62⋊13D4
►On 72 points
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 39 61 21 15 58)(2 40 62 22 16 59)(3 41 63 23 17 60)(4 42 64 24 18 55)(5 37 65 19 13 56)(6 38 66 20 14 57)(7 31 45 67 26 49)(8 32 46 68 27 50)(9 33 47 69 28 51)(10 34 48 70 29 52)(11 35 43 71 30 53)(12 36 44 72 25 54)
(1 46 24 43)(2 49 19 52)(3 44 20 47)(4 53 21 50)(5 48 22 45)(6 51 23 54)(7 65 10 59)(8 55 11 61)(9 63 12 57)(13 29 40 26)(14 33 41 36)(15 27 42 30)(16 31 37 34)(17 25 38 28)(18 35 39 32)(56 70 62 67)(58 68 64 71)(60 72 66 69)
(2 6)(3 5)(7 25)(8 30)(9 29)(10 28)(11 27)(12 26)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(19 23)(20 22)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 46)(44 45)(47 48)(49 54)(50 53)(51 52)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,39,61,21,15,58)(2,40,62,22,16,59)(3,41,63,23,17,60)(4,42,64,24,18,55)(5,37,65,19,13,56)(6,38,66,20,14,57)(7,31,45,67,26,49)(8,32,46,68,27,50)(9,33,47,69,28,51)(10,34,48,70,29,52)(11,35,43,71,30,53)(12,36,44,72,25,54), (1,46,24,43)(2,49,19,52)(3,44,20,47)(4,53,21,50)(5,48,22,45)(6,51,23,54)(7,65,10,59)(8,55,11,61)(9,63,12,57)(13,29,40,26)(14,33,41,36)(15,27,42,30)(16,31,37,34)(17,25,38,28)(18,35,39,32)(56,70,62,67)(58,68,64,71)(60,72,66,69), (2,6)(3,5)(7,25)(8,30)(9,29)(10,28)(11,27)(12,26)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,23)(20,22)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52)>;
G:=Group( (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,39,61,21,15,58)(2,40,62,22,16,59)(3,41,63,23,17,60)(4,42,64,24,18,55)(5,37,65,19,13,56)(6,38,66,20,14,57)(7,31,45,67,26,49)(8,32,46,68,27,50)(9,33,47,69,28,51)(10,34,48,70,29,52)(11,35,43,71,30,53)(12,36,44,72,25,54), (1,46,24,43)(2,49,19,52)(3,44,20,47)(4,53,21,50)(5,48,22,45)(6,51,23,54)(7,65,10,59)(8,55,11,61)(9,63,12,57)(13,29,40,26)(14,33,41,36)(15,27,42,30)(16,31,37,34)(17,25,38,28)(18,35,39,32)(56,70,62,67)(58,68,64,71)(60,72,66,69), (2,6)(3,5)(7,25)(8,30)(9,29)(10,28)(11,27)(12,26)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,23)(20,22)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52) );
G=PermutationGroup([[(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,39,61,21,15,58),(2,40,62,22,16,59),(3,41,63,23,17,60),(4,42,64,24,18,55),(5,37,65,19,13,56),(6,38,66,20,14,57),(7,31,45,67,26,49),(8,32,46,68,27,50),(9,33,47,69,28,51),(10,34,48,70,29,52),(11,35,43,71,30,53),(12,36,44,72,25,54)], [(1,46,24,43),(2,49,19,52),(3,44,20,47),(4,53,21,50),(5,48,22,45),(6,51,23,54),(7,65,10,59),(8,55,11,61),(9,63,12,57),(13,29,40,26),(14,33,41,36),(15,27,42,30),(16,31,37,34),(17,25,38,28),(18,35,39,32),(56,70,62,67),(58,68,64,71),(60,72,66,69)], [(2,6),(3,5),(7,25),(8,30),(9,29),(10,28),(11,27),(12,26),(13,63),(14,62),(15,61),(16,66),(17,65),(18,64),(19,23),(20,22),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,46),(44,45),(47,48),(49,54),(50,53),(51,52)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | S3×D4 |
| kernel | C62⋊13D4 | C6.11D12 | C62⋊5C4 | C2×C32⋊7D4 | D4×C3×C6 | C23×C3⋊S3 | C6×D4 | C2×C3⋊S3 | C62 | C2×C12 | C22×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 8 |
Matrix representation of C62⋊13D4 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,11,11,0,0,0,0,9,2,0,0,0,0,0,0,12,12,0,0,0,0,2,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;
C62⋊13D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{13}D_4 % in TeX
G:=Group("C6^2:13D4"); // GroupNames label
G:=SmallGroup(288,794);
// by ID
G=gap.SmallGroup(288,794);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations