metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.134D6, C6.142- 1+4, (C4×Q8)⋊20S3, C4⋊C4.301D6, (Q8×C12)⋊18C2, (C4×Dic6)⋊40C2, (C2×Q8).206D6, C4.50(C4○D12), Dic3.Q8⋊10C2, C42⋊2S3.5C2, C42⋊3S3.2C2, (C2×C6).127C24, C4.D12.10C2, D6⋊3Q8.10C2, C2.24(Q8○D12), C4.Dic6⋊17C2, C12.6Q8⋊27C2, Dic3⋊Q8⋊10C2, C12.121(C4○D4), (C2×C12).624C23, (C4×C12).179C22, D6⋊C4.126C22, (C6×Q8).227C22, Dic3⋊C4.78C22, (C22×S3).49C23, C4⋊Dic3.370C22, C22.148(S3×C23), (C2×Dic3).58C23, (C4×Dic3).87C22, C2.15(Q8.15D6), C3⋊2(C22.35C24), (C2×Dic6).292C22, C6.57(C2×C4○D4), C4⋊C4⋊S3.1C2, C2.66(C2×C4○D12), (S3×C2×C4).77C22, (C3×C4⋊C4).355C22, (C2×C4).290(C22×S3), SmallGroup(192,1142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.134D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c5 >
Subgroups: 408 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C6×Q8, C22.35C24, C4×Dic6, C12.6Q8, C42⋊2S3, C42⋊3S3, Dic3.Q8, C4.Dic6, C4.D12, C4⋊C4⋊S3, Dic3⋊Q8, D6⋊3Q8, Q8×C12, C42.134D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.35C24, C2×C4○D12, Q8.15D6, Q8○D12, C42.134D6
►On 96 points
(1 50 42 91)(2 92 43 51)(3 52 44 93)(4 94 45 53)(5 54 46 95)(6 96 47 55)(7 56 48 85)(8 86 37 57)(9 58 38 87)(10 88 39 59)(11 60 40 89)(12 90 41 49)(13 35 78 61)(14 62 79 36)(15 25 80 63)(16 64 81 26)(17 27 82 65)(18 66 83 28)(19 29 84 67)(20 68 73 30)(21 31 74 69)(22 70 75 32)(23 33 76 71)(24 72 77 34)
(1 61 48 29)(2 62 37 30)(3 63 38 31)(4 64 39 32)(5 65 40 33)(6 66 41 34)(7 67 42 35)(8 68 43 36)(9 69 44 25)(10 70 45 26)(11 71 46 27)(12 72 47 28)(13 85 84 50)(14 86 73 51)(15 87 74 52)(16 88 75 53)(17 89 76 54)(18 90 77 55)(19 91 78 56)(20 92 79 57)(21 93 80 58)(22 94 81 59)(23 95 82 60)(24 96 83 49)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 42 47)(2 46 43 5)(3 4 44 45)(7 12 48 41)(8 40 37 11)(9 10 38 39)(13 83 78 18)(14 17 79 82)(15 81 80 16)(19 77 84 24)(20 23 73 76)(21 75 74 22)(25 64 63 26)(27 62 65 36)(28 35 66 61)(29 72 67 34)(30 33 68 71)(31 70 69 32)(49 85 90 56)(50 55 91 96)(51 95 92 54)(52 53 93 94)(57 89 86 60)(58 59 87 88)
G:=sub<Sym(96)| (1,50,42,91)(2,92,43,51)(3,52,44,93)(4,94,45,53)(5,54,46,95)(6,96,47,55)(7,56,48,85)(8,86,37,57)(9,58,38,87)(10,88,39,59)(11,60,40,89)(12,90,41,49)(13,35,78,61)(14,62,79,36)(15,25,80,63)(16,64,81,26)(17,27,82,65)(18,66,83,28)(19,29,84,67)(20,68,73,30)(21,31,74,69)(22,70,75,32)(23,33,76,71)(24,72,77,34), (1,61,48,29)(2,62,37,30)(3,63,38,31)(4,64,39,32)(5,65,40,33)(6,66,41,34)(7,67,42,35)(8,68,43,36)(9,69,44,25)(10,70,45,26)(11,71,46,27)(12,72,47,28)(13,85,84,50)(14,86,73,51)(15,87,74,52)(16,88,75,53)(17,89,76,54)(18,90,77,55)(19,91,78,56)(20,92,79,57)(21,93,80,58)(22,94,81,59)(23,95,82,60)(24,96,83,49), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,42,47)(2,46,43,5)(3,4,44,45)(7,12,48,41)(8,40,37,11)(9,10,38,39)(13,83,78,18)(14,17,79,82)(15,81,80,16)(19,77,84,24)(20,23,73,76)(21,75,74,22)(25,64,63,26)(27,62,65,36)(28,35,66,61)(29,72,67,34)(30,33,68,71)(31,70,69,32)(49,85,90,56)(50,55,91,96)(51,95,92,54)(52,53,93,94)(57,89,86,60)(58,59,87,88)>;
G:=Group( (1,50,42,91)(2,92,43,51)(3,52,44,93)(4,94,45,53)(5,54,46,95)(6,96,47,55)(7,56,48,85)(8,86,37,57)(9,58,38,87)(10,88,39,59)(11,60,40,89)(12,90,41,49)(13,35,78,61)(14,62,79,36)(15,25,80,63)(16,64,81,26)(17,27,82,65)(18,66,83,28)(19,29,84,67)(20,68,73,30)(21,31,74,69)(22,70,75,32)(23,33,76,71)(24,72,77,34), (1,61,48,29)(2,62,37,30)(3,63,38,31)(4,64,39,32)(5,65,40,33)(6,66,41,34)(7,67,42,35)(8,68,43,36)(9,69,44,25)(10,70,45,26)(11,71,46,27)(12,72,47,28)(13,85,84,50)(14,86,73,51)(15,87,74,52)(16,88,75,53)(17,89,76,54)(18,90,77,55)(19,91,78,56)(20,92,79,57)(21,93,80,58)(22,94,81,59)(23,95,82,60)(24,96,83,49), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,42,47)(2,46,43,5)(3,4,44,45)(7,12,48,41)(8,40,37,11)(9,10,38,39)(13,83,78,18)(14,17,79,82)(15,81,80,16)(19,77,84,24)(20,23,73,76)(21,75,74,22)(25,64,63,26)(27,62,65,36)(28,35,66,61)(29,72,67,34)(30,33,68,71)(31,70,69,32)(49,85,90,56)(50,55,91,96)(51,95,92,54)(52,53,93,94)(57,89,86,60)(58,59,87,88) );
G=PermutationGroup([[(1,50,42,91),(2,92,43,51),(3,52,44,93),(4,94,45,53),(5,54,46,95),(6,96,47,55),(7,56,48,85),(8,86,37,57),(9,58,38,87),(10,88,39,59),(11,60,40,89),(12,90,41,49),(13,35,78,61),(14,62,79,36),(15,25,80,63),(16,64,81,26),(17,27,82,65),(18,66,83,28),(19,29,84,67),(20,68,73,30),(21,31,74,69),(22,70,75,32),(23,33,76,71),(24,72,77,34)], [(1,61,48,29),(2,62,37,30),(3,63,38,31),(4,64,39,32),(5,65,40,33),(6,66,41,34),(7,67,42,35),(8,68,43,36),(9,69,44,25),(10,70,45,26),(11,71,46,27),(12,72,47,28),(13,85,84,50),(14,86,73,51),(15,87,74,52),(16,88,75,53),(17,89,76,54),(18,90,77,55),(19,91,78,56),(20,92,79,57),(21,93,80,58),(22,94,81,59),(23,95,82,60),(24,96,83,49)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,42,47),(2,46,43,5),(3,4,44,45),(7,12,48,41),(8,40,37,11),(9,10,38,39),(13,83,78,18),(14,17,79,82),(15,81,80,16),(19,77,84,24),(20,23,73,76),(21,75,74,22),(25,64,63,26),(27,62,65,36),(28,35,66,61),(29,72,67,34),(30,33,68,71),(31,70,69,32),(49,85,90,56),(50,55,91,96),(51,95,92,54),(52,53,93,94),(57,89,86,60),(58,59,87,88)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- 1+4 | Q8.15D6 | Q8○D12 |
| kernel | C42.134D6 | C4×Dic6 | C12.6Q8 | C42⋊2S3 | C42⋊3S3 | Dic3.Q8 | C4.Dic6 | C4.D12 | C4⋊C4⋊S3 | Dic3⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 8 | 2 | 2 | 2 |
Matrix representation of C42.134D6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 1 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 5 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 | 5 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 | 0 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,10,1,8,0,0,0,0,0,0,5,0,8],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,1,5,0,0,0,0,0,12,0,2,0,0,0,0,11,5,1,0,0,0,0,0,0,12,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,5,1,8,0,0,0,0,0,1,0,11,0,0,0,0,11,5,1,0,0,0,0,0,0,1,0,12] >;
C42.134D6 in GAP, Magma, Sage, TeX
C_4^2._{134}D_6 % in TeX
G:=Group("C4^2.134D6"); // GroupNames label
G:=SmallGroup(192,1142);
// by ID
G=gap.SmallGroup(192,1142);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations