metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.137D6, C6.872- 1+4, C6.702+ 1+4, C4.4D4⋊6S3, C42⋊3S3⋊7C2, D6⋊3Q8⋊27C2, (C4×Dic6)⋊43C2, (C2×D4).107D6, (C2×Q8).105D6, C22⋊C4.71D6, C23.9D6⋊40C2, (C2×C6).213C24, D6⋊C4.59C22, C2.48(Q8○D12), Dic3⋊4D4⋊28C2, Dic3⋊Q8⋊20C2, C2.72(D4⋊6D6), (C2×C12).629C23, (C4×C12).183C22, C23.14D6.5C2, (C6×D4).207C22, C23.8D6⋊36C2, (C22×C6).43C23, C23.45(C22×S3), (C6×Q8).122C22, Dic3.11(C4○D4), C23.11D6⋊37C2, Dic3.D4⋊37C2, C23.23D6⋊24C2, C23.16D6⋊16C2, Dic3⋊C4.82C22, (C22×S3).93C23, C4⋊Dic3.232C22, C22.234(S3×C23), C3⋊8(C22.36C24), (C4×Dic3).213C22, (C2×Dic3).250C23, (C2×Dic6).174C22, C6.D4.50C22, (C22×Dic3).138C22, C2.72(S3×C4○D4), C6.184(C2×C4○D4), (C3×C4.4D4)⋊7C2, (S3×C2×C4).119C22, (C2×C4).191(C22×S3), (C2×C3⋊D4).56C22, (C3×C22⋊C4).60C22, SmallGroup(192,1228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.137D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 528 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C22.36C24, C4×Dic6, C42⋊3S3, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6, C23.11D6, C23.23D6, C23.14D6, Dic3⋊Q8, D6⋊3Q8, C3×C4.4D4, C42.137D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, S3×C23, C22.36C24, D4⋊6D6, S3×C4○D4, Q8○D12, C42.137D6
►On 96 points
(1 67 55 23)(2 48 56 86)(3 69 57 19)(4 44 58 88)(5 71 59 21)(6 46 60 90)(7 61 29 17)(8 42 30 80)(9 63 25 13)(10 38 26 82)(11 65 27 15)(12 40 28 84)(14 54 64 94)(16 50 66 96)(18 52 62 92)(20 76 70 32)(22 78 72 34)(24 74 68 36)(31 87 75 43)(33 89 77 45)(35 85 73 47)(37 93 81 53)(39 95 83 49)(41 91 79 51)
(1 79 73 17)(2 42 74 62)(3 81 75 13)(4 38 76 64)(5 83 77 15)(6 40 78 66)(7 67 51 47)(8 24 52 86)(9 69 53 43)(10 20 54 88)(11 71 49 45)(12 22 50 90)(14 58 82 32)(16 60 84 34)(18 56 80 36)(19 93 87 25)(21 95 89 27)(23 91 85 29)(26 70 94 44)(28 72 96 46)(30 68 92 48)(31 63 57 37)(33 65 59 39)(35 61 55 41)
(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 25 73 93)(2 30 74 92)(3 29 75 91)(4 28 76 96)(5 27 77 95)(6 26 78 94)(7 31 51 57)(8 36 52 56)(9 35 53 55)(10 34 54 60)(11 33 49 59)(12 32 50 58)(13 67 81 47)(14 72 82 46)(15 71 83 45)(16 70 84 44)(17 69 79 43)(18 68 80 48)(19 41 87 61)(20 40 88 66)(21 39 89 65)(22 38 90 64)(23 37 85 63)(24 42 86 62)
G:=sub<Sym(96)| (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,42,74,62)(3,81,75,13)(4,38,76,64)(5,83,77,15)(6,40,78,66)(7,67,51,47)(8,24,52,86)(9,69,53,43)(10,20,54,88)(11,71,49,45)(12,22,50,90)(14,58,82,32)(16,60,84,34)(18,56,80,36)(19,93,87,25)(21,95,89,27)(23,91,85,29)(26,70,94,44)(28,72,96,46)(30,68,92,48)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,25,73,93)(2,30,74,92)(3,29,75,91)(4,28,76,96)(5,27,77,95)(6,26,78,94)(7,31,51,57)(8,36,52,56)(9,35,53,55)(10,34,54,60)(11,33,49,59)(12,32,50,58)(13,67,81,47)(14,72,82,46)(15,71,83,45)(16,70,84,44)(17,69,79,43)(18,68,80,48)(19,41,87,61)(20,40,88,66)(21,39,89,65)(22,38,90,64)(23,37,85,63)(24,42,86,62)>;
G:=Group( (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,42,74,62)(3,81,75,13)(4,38,76,64)(5,83,77,15)(6,40,78,66)(7,67,51,47)(8,24,52,86)(9,69,53,43)(10,20,54,88)(11,71,49,45)(12,22,50,90)(14,58,82,32)(16,60,84,34)(18,56,80,36)(19,93,87,25)(21,95,89,27)(23,91,85,29)(26,70,94,44)(28,72,96,46)(30,68,92,48)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,25,73,93)(2,30,74,92)(3,29,75,91)(4,28,76,96)(5,27,77,95)(6,26,78,94)(7,31,51,57)(8,36,52,56)(9,35,53,55)(10,34,54,60)(11,33,49,59)(12,32,50,58)(13,67,81,47)(14,72,82,46)(15,71,83,45)(16,70,84,44)(17,69,79,43)(18,68,80,48)(19,41,87,61)(20,40,88,66)(21,39,89,65)(22,38,90,64)(23,37,85,63)(24,42,86,62) );
G=PermutationGroup([[(1,67,55,23),(2,48,56,86),(3,69,57,19),(4,44,58,88),(5,71,59,21),(6,46,60,90),(7,61,29,17),(8,42,30,80),(9,63,25,13),(10,38,26,82),(11,65,27,15),(12,40,28,84),(14,54,64,94),(16,50,66,96),(18,52,62,92),(20,76,70,32),(22,78,72,34),(24,74,68,36),(31,87,75,43),(33,89,77,45),(35,85,73,47),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(1,79,73,17),(2,42,74,62),(3,81,75,13),(4,38,76,64),(5,83,77,15),(6,40,78,66),(7,67,51,47),(8,24,52,86),(9,69,53,43),(10,20,54,88),(11,71,49,45),(12,22,50,90),(14,58,82,32),(16,60,84,34),(18,56,80,36),(19,93,87,25),(21,95,89,27),(23,91,85,29),(26,70,94,44),(28,72,96,46),(30,68,92,48),(31,63,57,37),(33,65,59,39),(35,61,55,41)], [(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)], [(1,25,73,93),(2,30,74,92),(3,29,75,91),(4,28,76,96),(5,27,77,95),(6,26,78,94),(7,31,51,57),(8,36,52,56),(9,35,53,55),(10,34,54,60),(11,33,49,59),(12,32,50,58),(13,67,81,47),(14,72,82,46),(15,71,83,45),(16,70,84,44),(17,69,79,43),(18,68,80,48),(19,41,87,61),(20,40,88,66),(21,39,89,65),(22,38,90,64),(23,37,85,63),(24,42,86,62)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | S3×C4○D4 | Q8○D12 |
| kernel | C42.137D6 | C4×Dic6 | C42⋊3S3 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | C23.23D6 | C23.14D6 | Dic3⋊Q8 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.137D6 ►in GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 | 1 |
| 2 | 12 | 0 | 10 | 0 | 0 | 0 | 0 |
| 1 | 1 | 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 3 | 11 | 1 | 0 | 0 | 0 | 0 |
| 10 | 3 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 5 | 9 |
| 0 | 0 | 0 | 0 | 11 | 5 | 6 | 8 |
| 0 | 0 | 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 | 1 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,0,1,0,1,0,0,0,0,1,12,0,0,0,0,0,0,0,11,0,12],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,1,12,0,0,0,0,1,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1],[2,1,0,10,0,0,0,0,12,1,3,3,0,0,0,0,0,3,11,12,0,0,0,0,10,10,1,12,0,0,0,0,0,0,0,0,6,11,2,11,0,0,0,0,11,7,2,5,0,0,0,0,0,0,5,6,0,0,0,0,0,0,9,8],[0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,12,0,12,0,0,0,0,0,1,1,0,1,0,0,0,0,2,0,0,1] >;
C42.137D6 in GAP, Magma, Sage, TeX
C_4^2._{137}D_6 % in TeX
G:=Group("C4^2.137D6"); // GroupNames label
G:=SmallGroup(192,1228);
// by ID
G=gap.SmallGroup(192,1228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,100,1123,346,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations