metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.56D6, (C2×D4).36D6, C4⋊D4.4S3, C6.95(C4○D8), C6.Q16⋊35C2, (C2×C12).261D4, (C22×C6).81D4, C12.55D4⋊9C2, D4⋊Dic3⋊14C2, C6.89(C8⋊C22), C12.Q8⋊34C2, (C6×D4).52C22, (C22×C4).134D6, C12.182(C4○D4), C4.92(D4⋊2S3), (C2×C12).354C23, C3⋊7(C23.19D4), C23.29(C3⋊D4), C2.14(Q8.13D6), C2.10(D12⋊6C22), C23.26D6⋊15C2, C4⋊Dic3.336C22, (C22×C12).158C22, C6.79(C22.D4), C2.13(C23.23D6), (C3×C4⋊D4).3C2, (C2×C6).485(C2×D4), (C2×C3⋊C8).107C22, (C2×C4).170(C3⋊D4), (C3×C4⋊C4).103C22, (C2×C4).454(C22×S3), C22.160(C2×C3⋊D4), SmallGroup(192,594)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Q16⋊C2
G = < a,b,c,d | a12=b4=d2=1, c2=a9b2, bab-1=dad=a7, cac-1=a5, cbc-1=a9b-1, dbd=a6b-1, dcd=a9c >
Subgroups: 272 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C23.19D4, C6.Q16, C12.Q8, C12.55D4, D4⋊Dic3, C23.26D6, C3×C4⋊D4, C6.Q16⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C4○D8, C8⋊C22, D4⋊2S3, C2×C3⋊D4, C23.19D4, D12⋊6C22, C23.23D6, Q8.13D6, C6.Q16⋊C2
►On 96 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)(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 51 80)(2 32 52 75)(3 27 53 82)(4 34 54 77)(5 29 55 84)(6 36 56 79)(7 31 57 74)(8 26 58 81)(9 33 59 76)(10 28 60 83)(11 35 49 78)(12 30 50 73)(13 39 71 88)(14 46 72 95)(15 41 61 90)(16 48 62 85)(17 43 63 92)(18 38 64 87)(19 45 65 94)(20 40 66 89)(21 47 67 96)(22 42 68 91)(23 37 69 86)(24 44 70 93)
(1 68 60 19 7 62 54 13)(2 61 49 24 8 67 55 18)(3 66 50 17 9 72 56 23)(4 71 51 22 10 65 57 16)(5 64 52 15 11 70 58 21)(6 69 53 20 12 63 59 14)(25 39 77 85 31 45 83 91)(26 44 78 90 32 38 84 96)(27 37 79 95 33 43 73 89)(28 42 80 88 34 48 74 94)(29 47 81 93 35 41 75 87)(30 40 82 86 36 46 76 92)
(2 8)(4 10)(6 12)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)(25 74)(26 81)(27 76)(28 83)(29 78)(30 73)(31 80)(32 75)(33 82)(34 77)(35 84)(36 79)(37 89)(38 96)(39 91)(40 86)(41 93)(42 88)(43 95)(44 90)(45 85)(46 92)(47 87)(48 94)(50 56)(52 58)(54 60)(61 64)(62 71)(63 66)(65 68)(67 70)(69 72)
G:=sub<Sym(96)| (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), (1,25,51,80)(2,32,52,75)(3,27,53,82)(4,34,54,77)(5,29,55,84)(6,36,56,79)(7,31,57,74)(8,26,58,81)(9,33,59,76)(10,28,60,83)(11,35,49,78)(12,30,50,73)(13,39,71,88)(14,46,72,95)(15,41,61,90)(16,48,62,85)(17,43,63,92)(18,38,64,87)(19,45,65,94)(20,40,66,89)(21,47,67,96)(22,42,68,91)(23,37,69,86)(24,44,70,93), (1,68,60,19,7,62,54,13)(2,61,49,24,8,67,55,18)(3,66,50,17,9,72,56,23)(4,71,51,22,10,65,57,16)(5,64,52,15,11,70,58,21)(6,69,53,20,12,63,59,14)(25,39,77,85,31,45,83,91)(26,44,78,90,32,38,84,96)(27,37,79,95,33,43,73,89)(28,42,80,88,34,48,74,94)(29,47,81,93,35,41,75,87)(30,40,82,86,36,46,76,92), (2,8)(4,10)(6,12)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(25,74)(26,81)(27,76)(28,83)(29,78)(30,73)(31,80)(32,75)(33,82)(34,77)(35,84)(36,79)(37,89)(38,96)(39,91)(40,86)(41,93)(42,88)(43,95)(44,90)(45,85)(46,92)(47,87)(48,94)(50,56)(52,58)(54,60)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,25,51,80)(2,32,52,75)(3,27,53,82)(4,34,54,77)(5,29,55,84)(6,36,56,79)(7,31,57,74)(8,26,58,81)(9,33,59,76)(10,28,60,83)(11,35,49,78)(12,30,50,73)(13,39,71,88)(14,46,72,95)(15,41,61,90)(16,48,62,85)(17,43,63,92)(18,38,64,87)(19,45,65,94)(20,40,66,89)(21,47,67,96)(22,42,68,91)(23,37,69,86)(24,44,70,93), (1,68,60,19,7,62,54,13)(2,61,49,24,8,67,55,18)(3,66,50,17,9,72,56,23)(4,71,51,22,10,65,57,16)(5,64,52,15,11,70,58,21)(6,69,53,20,12,63,59,14)(25,39,77,85,31,45,83,91)(26,44,78,90,32,38,84,96)(27,37,79,95,33,43,73,89)(28,42,80,88,34,48,74,94)(29,47,81,93,35,41,75,87)(30,40,82,86,36,46,76,92), (2,8)(4,10)(6,12)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(25,74)(26,81)(27,76)(28,83)(29,78)(30,73)(31,80)(32,75)(33,82)(34,77)(35,84)(36,79)(37,89)(38,96)(39,91)(40,86)(41,93)(42,88)(43,95)(44,90)(45,85)(46,92)(47,87)(48,94)(50,56)(52,58)(54,60)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72) );
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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,25,51,80),(2,32,52,75),(3,27,53,82),(4,34,54,77),(5,29,55,84),(6,36,56,79),(7,31,57,74),(8,26,58,81),(9,33,59,76),(10,28,60,83),(11,35,49,78),(12,30,50,73),(13,39,71,88),(14,46,72,95),(15,41,61,90),(16,48,62,85),(17,43,63,92),(18,38,64,87),(19,45,65,94),(20,40,66,89),(21,47,67,96),(22,42,68,91),(23,37,69,86),(24,44,70,93)], [(1,68,60,19,7,62,54,13),(2,61,49,24,8,67,55,18),(3,66,50,17,9,72,56,23),(4,71,51,22,10,65,57,16),(5,64,52,15,11,70,58,21),(6,69,53,20,12,63,59,14),(25,39,77,85,31,45,83,91),(26,44,78,90,32,38,84,96),(27,37,79,95,33,43,73,89),(28,42,80,88,34,48,74,94),(29,47,81,93,35,41,75,87),(30,40,82,86,36,46,76,92)], [(2,8),(4,10),(6,12),(13,16),(14,23),(15,18),(17,20),(19,22),(21,24),(25,74),(26,81),(27,76),(28,83),(29,78),(30,73),(31,80),(32,75),(33,82),(34,77),(35,84),(36,79),(37,89),(38,96),(39,91),(40,86),(41,93),(42,88),(43,95),(44,90),(45,85),(46,92),(47,87),(48,94),(50,56),(52,58),(54,60),(61,64),(62,71),(63,66),(65,68),(67,70),(69,72)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | C4○D8 | C8⋊C22 | D4⋊2S3 | D12⋊6C22 | Q8.13D6 |
| kernel | C6.Q16⋊C2 | C6.Q16 | C12.Q8 | C12.55D4 | D4⋊Dic3 | C23.26D6 | C3×C4⋊D4 | C4⋊D4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C4 | C23 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C6.Q16⋊C2 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 56 | 3 | 0 | 0 | 0 | 0 |
| 25 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 59 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 3 |
| 0 | 0 | 0 | 0 | 31 | 45 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[56,25,0,0,0,0,3,17,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,59,0,0,0,0,0,27,0,0,0,0,0,0,57,57,0,0,0,0,16,57,0,0,0,0,0,0,28,31,0,0,0,0,3,45],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C6.Q16⋊C2 in GAP, Magma, Sage, TeX
C_6.Q_{16}\rtimes C_2 % in TeX
G:=Group("C6.Q16:C2"); // GroupNames label
G:=SmallGroup(192,594);
// by ID
G=gap.SmallGroup(192,594);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=a^9*b^2,b*a*b^-1=d*a*d=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1,d*b*d=a^6*b^-1,d*c*d=a^9*c>;
// generators/relations