metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.236D6, C42⋊C2⋊3S3, C6.86(C4○D8), C6.D8⋊28C2, (C2×C12).450D4, C6.Q16⋊29C2, (C22×C6).76D4, C4.91(C4○D12), C12⋊7D4.11C2, C12.55D4⋊6C2, C2.6(D4⋊D6), C12.Q8⋊29C2, (C22×C4).127D6, C12.179(C4○D4), C6.106(C8⋊C22), (C2×C12).330C23, C2.9(Q8.13D6), (C2×D12).92C22, C3⋊6(C23.19D4), C23.27(C3⋊D4), C4⋊Dic3.135C22, (C22×C12).152C22, C6.66(C22.D4), C2.17(C23.28D6), (C2×C6).459(C2×D4), (C2×C3⋊C8).87C22, (C3×C42⋊C2)⋊3C2, (C2×C4).215(C3⋊D4), (C3×C4⋊C4).267C22, (C2×C4).430(C22×S3), C22.145(C2×C3⋊D4), SmallGroup(192,562)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.236D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c5 >
Subgroups: 328 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C23.19D4, C6.Q16, C12.Q8, C6.D8, C12.55D4, C12⋊7D4, C3×C42⋊C2, C4⋊C4.236D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C4○D8, C8⋊C22, C4○D12, C2×C3⋊D4, C23.19D4, C23.28D6, D4⋊D6, Q8.13D6, C4⋊C4.236D6
►On 96 points
(1 60 7 54)(2 49 8 55)(3 50 9 56)(4 51 10 57)(5 52 11 58)(6 53 12 59)(13 72 19 66)(14 61 20 67)(15 62 21 68)(16 63 22 69)(17 64 23 70)(18 65 24 71)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 75 43 81)(38 76 44 82)(39 77 45 83)(40 78 46 84)(41 79 47 73)(42 80 48 74)
(1 85 18 78)(2 86 19 79)(3 87 20 80)(4 88 21 81)(5 89 22 82)(6 90 23 83)(7 91 24 84)(8 92 13 73)(9 93 14 74)(10 94 15 75)(11 95 16 76)(12 96 17 77)(25 71 46 54)(26 72 47 55)(27 61 48 56)(28 62 37 57)(29 63 38 58)(30 64 39 59)(31 65 40 60)(32 66 41 49)(33 67 42 50)(34 68 43 51)(35 69 44 52)(36 70 45 53)
(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 18 23)(2 22 19 5)(3 4 20 21)(7 12 24 17)(8 16 13 11)(9 10 14 15)(25 96 46 77)(26 76 47 95)(27 94 48 75)(28 74 37 93)(29 92 38 73)(30 84 39 91)(31 90 40 83)(32 82 41 89)(33 88 42 81)(34 80 43 87)(35 86 44 79)(36 78 45 85)(49 63 66 58)(50 57 67 62)(51 61 68 56)(52 55 69 72)(53 71 70 54)(59 65 64 60)
G:=sub<Sym(96)| (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,72,19,66)(14,61,20,67)(15,62,21,68)(16,63,22,69)(17,64,23,70)(18,65,24,71)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74), (1,85,18,78)(2,86,19,79)(3,87,20,80)(4,88,21,81)(5,89,22,82)(6,90,23,83)(7,91,24,84)(8,92,13,73)(9,93,14,74)(10,94,15,75)(11,95,16,76)(12,96,17,77)(25,71,46,54)(26,72,47,55)(27,61,48,56)(28,62,37,57)(29,63,38,58)(30,64,39,59)(31,65,40,60)(32,66,41,49)(33,67,42,50)(34,68,43,51)(35,69,44,52)(36,70,45,53), (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), (1,6,18,23)(2,22,19,5)(3,4,20,21)(7,12,24,17)(8,16,13,11)(9,10,14,15)(25,96,46,77)(26,76,47,95)(27,94,48,75)(28,74,37,93)(29,92,38,73)(30,84,39,91)(31,90,40,83)(32,82,41,89)(33,88,42,81)(34,80,43,87)(35,86,44,79)(36,78,45,85)(49,63,66,58)(50,57,67,62)(51,61,68,56)(52,55,69,72)(53,71,70,54)(59,65,64,60)>;
G:=Group( (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,72,19,66)(14,61,20,67)(15,62,21,68)(16,63,22,69)(17,64,23,70)(18,65,24,71)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74), (1,85,18,78)(2,86,19,79)(3,87,20,80)(4,88,21,81)(5,89,22,82)(6,90,23,83)(7,91,24,84)(8,92,13,73)(9,93,14,74)(10,94,15,75)(11,95,16,76)(12,96,17,77)(25,71,46,54)(26,72,47,55)(27,61,48,56)(28,62,37,57)(29,63,38,58)(30,64,39,59)(31,65,40,60)(32,66,41,49)(33,67,42,50)(34,68,43,51)(35,69,44,52)(36,70,45,53), (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), (1,6,18,23)(2,22,19,5)(3,4,20,21)(7,12,24,17)(8,16,13,11)(9,10,14,15)(25,96,46,77)(26,76,47,95)(27,94,48,75)(28,74,37,93)(29,92,38,73)(30,84,39,91)(31,90,40,83)(32,82,41,89)(33,88,42,81)(34,80,43,87)(35,86,44,79)(36,78,45,85)(49,63,66,58)(50,57,67,62)(51,61,68,56)(52,55,69,72)(53,71,70,54)(59,65,64,60) );
G=PermutationGroup([[(1,60,7,54),(2,49,8,55),(3,50,9,56),(4,51,10,57),(5,52,11,58),(6,53,12,59),(13,72,19,66),(14,61,20,67),(15,62,21,68),(16,63,22,69),(17,64,23,70),(18,65,24,71),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,75,43,81),(38,76,44,82),(39,77,45,83),(40,78,46,84),(41,79,47,73),(42,80,48,74)], [(1,85,18,78),(2,86,19,79),(3,87,20,80),(4,88,21,81),(5,89,22,82),(6,90,23,83),(7,91,24,84),(8,92,13,73),(9,93,14,74),(10,94,15,75),(11,95,16,76),(12,96,17,77),(25,71,46,54),(26,72,47,55),(27,61,48,56),(28,62,37,57),(29,63,38,58),(30,64,39,59),(31,65,40,60),(32,66,41,49),(33,67,42,50),(34,68,43,51),(35,69,44,52),(36,70,45,53)], [(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)], [(1,6,18,23),(2,22,19,5),(3,4,20,21),(7,12,24,17),(8,16,13,11),(9,10,14,15),(25,96,46,77),(26,76,47,95),(27,94,48,75),(28,74,37,93),(29,92,38,73),(30,84,39,91),(31,90,40,83),(32,82,41,89),(33,88,42,81),(34,80,43,87),(35,86,44,79),(36,78,45,85),(49,63,66,58),(50,57,67,62),(51,61,68,56),(52,55,69,72),(53,71,70,54),(59,65,64,60)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | C4○D8 | C4○D12 | C8⋊C22 | D4⋊D6 | Q8.13D6 |
| kernel | C4⋊C4.236D6 | C6.Q16 | C12.Q8 | C6.D8 | C12.55D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C12 | C2×C4 | C23 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.236D6 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 70 |
| 0 | 0 | 25 | 1 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 61 | 55 |
| 0 | 0 | 4 | 12 |
| 30 | 43 | 0 | 0 |
| 30 | 60 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 43 | 30 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 55 | 46 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,25,0,0,70,1],[7,59,0,0,14,66,0,0,0,0,61,4,0,0,55,12],[30,30,0,0,43,60,0,0,0,0,27,0,0,0,0,27],[43,60,0,0,30,30,0,0,0,0,27,55,0,0,0,46] >;
C4⋊C4.236D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{236}D_6 % in TeX
G:=Group("C4:C4.236D6"); // GroupNames label
G:=SmallGroup(192,562);
// by ID
G=gap.SmallGroup(192,562);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations