metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.225D6, C4.Dic3⋊7C4, C12.16(C4⋊C4), (C2×C12).16Q8, C12.63(C2×Q8), (C2×C12).132D4, C6.Q16⋊24C2, C4.28(C2×Dic6), (C2×C4).11Dic6, C6.82(C8⋊C22), C12.58(C22×C4), C12.Q8⋊24C2, (C22×C6).182D4, (C22×C4).108D6, C3⋊3(M4(2)⋊C4), C4.16(Dic3⋊C4), (C2×C12).317C23, C2.1(D12⋊6C22), C6.82(C8.C22), C23.82(C3⋊D4), C2.1(Q8.11D6), C4⋊Dic3.323C22, C22.15(Dic3⋊C4), (C22×C12).132C22, C23.26D6.12C2, C3⋊C8⋊6(C2×C4), C4.86(S3×C2×C4), (C6×C4⋊C4).5C2, C6.34(C2×C4⋊C4), (C2×C4⋊C4).6S3, (C2×C4).38(C4×S3), (C2×C6).46(C4⋊C4), (C2×C12).76(C2×C4), (C2×C6).437(C2×D4), (C2×C3⋊C8).79C22, C2.9(C2×Dic3⋊C4), C22.56(C2×C3⋊D4), (C2×C4).181(C3⋊D4), (C3×C4⋊C4).256C22, (C2×C4).417(C22×S3), (C2×C4.Dic3).15C2, SmallGroup(192,523)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.225D6
G = < a,b,c,d | a4=b4=c6=1, d2=a-1b2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b-1, dcd-1=a2c-1 >
Subgroups: 248 in 118 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, M4(2)⋊C4, C6.Q16, C12.Q8, C2×C4.Dic3, C23.26D6, C6×C4⋊C4, C4⋊C4.225D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2)⋊C4, C2×Dic3⋊C4, D12⋊6C22, Q8.11D6, C4⋊C4.225D6
►On 96 points
(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 75 13 79)(10 76 14 80)(11 77 15 73)(12 78 16 74)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)(49 84 53 88)(50 85 54 81)(51 86 55 82)(52 87 56 83)(65 89 69 93)(66 90 70 94)(67 91 71 95)(68 92 72 96)
(1 37 19 62)(2 36 20 61)(3 35 21 60)(4 34 22 59)(5 33 23 58)(6 40 24 57)(7 39 17 64)(8 38 18 63)(9 42 77 26)(10 41 78 25)(11 48 79 32)(12 47 80 31)(13 46 73 30)(14 45 74 29)(15 44 75 28)(16 43 76 27)(49 89 86 71)(50 96 87 70)(51 95 88 69)(52 94 81 68)(53 93 82 67)(54 92 83 66)(55 91 84 65)(56 90 85 72)
(1 65 28)(2 25 66 6 29 70)(3 67 30)(4 27 68 8 31 72)(5 69 32)(7 71 26)(9 39 49)(10 54 40 14 50 36)(11 33 51)(12 56 34 16 52 38)(13 35 53)(15 37 55)(17 89 42)(18 47 90 22 43 94)(19 91 44)(20 41 92 24 45 96)(21 93 46)(23 95 48)(57 74 87 61 78 83)(58 88 79)(59 76 81 63 80 85)(60 82 73)(62 84 75)(64 86 77)
(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)
G:=sub<Sym(96)| (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,75,13,79)(10,76,14,80)(11,77,15,73)(12,78,16,74)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,89,69,93)(66,90,70,94)(67,91,71,95)(68,92,72,96), (1,37,19,62)(2,36,20,61)(3,35,21,60)(4,34,22,59)(5,33,23,58)(6,40,24,57)(7,39,17,64)(8,38,18,63)(9,42,77,26)(10,41,78,25)(11,48,79,32)(12,47,80,31)(13,46,73,30)(14,45,74,29)(15,44,75,28)(16,43,76,27)(49,89,86,71)(50,96,87,70)(51,95,88,69)(52,94,81,68)(53,93,82,67)(54,92,83,66)(55,91,84,65)(56,90,85,72), (1,65,28)(2,25,66,6,29,70)(3,67,30)(4,27,68,8,31,72)(5,69,32)(7,71,26)(9,39,49)(10,54,40,14,50,36)(11,33,51)(12,56,34,16,52,38)(13,35,53)(15,37,55)(17,89,42)(18,47,90,22,43,94)(19,91,44)(20,41,92,24,45,96)(21,93,46)(23,95,48)(57,74,87,61,78,83)(58,88,79)(59,76,81,63,80,85)(60,82,73)(62,84,75)(64,86,77), (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)>;
G:=Group( (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,75,13,79)(10,76,14,80)(11,77,15,73)(12,78,16,74)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,89,69,93)(66,90,70,94)(67,91,71,95)(68,92,72,96), (1,37,19,62)(2,36,20,61)(3,35,21,60)(4,34,22,59)(5,33,23,58)(6,40,24,57)(7,39,17,64)(8,38,18,63)(9,42,77,26)(10,41,78,25)(11,48,79,32)(12,47,80,31)(13,46,73,30)(14,45,74,29)(15,44,75,28)(16,43,76,27)(49,89,86,71)(50,96,87,70)(51,95,88,69)(52,94,81,68)(53,93,82,67)(54,92,83,66)(55,91,84,65)(56,90,85,72), (1,65,28)(2,25,66,6,29,70)(3,67,30)(4,27,68,8,31,72)(5,69,32)(7,71,26)(9,39,49)(10,54,40,14,50,36)(11,33,51)(12,56,34,16,52,38)(13,35,53)(15,37,55)(17,89,42)(18,47,90,22,43,94)(19,91,44)(20,41,92,24,45,96)(21,93,46)(23,95,48)(57,74,87,61,78,83)(58,88,79)(59,76,81,63,80,85)(60,82,73)(62,84,75)(64,86,77), (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) );
G=PermutationGroup([[(1,17,5,21),(2,18,6,22),(3,19,7,23),(4,20,8,24),(9,75,13,79),(10,76,14,80),(11,77,15,73),(12,78,16,74),(25,47,29,43),(26,48,30,44),(27,41,31,45),(28,42,32,46),(33,64,37,60),(34,57,38,61),(35,58,39,62),(36,59,40,63),(49,84,53,88),(50,85,54,81),(51,86,55,82),(52,87,56,83),(65,89,69,93),(66,90,70,94),(67,91,71,95),(68,92,72,96)], [(1,37,19,62),(2,36,20,61),(3,35,21,60),(4,34,22,59),(5,33,23,58),(6,40,24,57),(7,39,17,64),(8,38,18,63),(9,42,77,26),(10,41,78,25),(11,48,79,32),(12,47,80,31),(13,46,73,30),(14,45,74,29),(15,44,75,28),(16,43,76,27),(49,89,86,71),(50,96,87,70),(51,95,88,69),(52,94,81,68),(53,93,82,67),(54,92,83,66),(55,91,84,65),(56,90,85,72)], [(1,65,28),(2,25,66,6,29,70),(3,67,30),(4,27,68,8,31,72),(5,69,32),(7,71,26),(9,39,49),(10,54,40,14,50,36),(11,33,51),(12,56,34,16,52,38),(13,35,53),(15,37,55),(17,89,42),(18,47,90,22,43,94),(19,91,44),(20,41,92,24,45,96),(21,93,46),(23,95,48),(57,74,87,61,78,83),(58,88,79),(59,76,81,63,80,85),(60,82,73),(62,84,75),(64,86,77)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
42 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 | C4 | S3 | D4 | Q8 | D4 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 | C3⋊D4 | C8⋊C22 | C8.C22 | D12⋊6C22 | Q8.11D6 |
| kernel | C4⋊C4.225D6 | C6.Q16 | C12.Q8 | C2×C4.Dic3 | C23.26D6 | C6×C4⋊C4 | C4.Dic3 | C2×C4⋊C4 | C2×C12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.225D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 66 | 0 | 0 |
| 0 | 0 | 42 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 66 |
| 0 | 0 | 0 | 0 | 42 | 1 |
| 52 | 8 | 0 | 0 | 0 | 0 |
| 36 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 15 | 0 | 0 |
| 0 | 0 | 65 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 50 |
| 0 | 0 | 0 | 0 | 17 | 68 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 0 | 0 | 65 |
| 13 | 67 | 0 | 0 | 0 | 0 |
| 4 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 9 | 63 | 0 | 0 |
| 0 | 0 | 60 | 64 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,42,0,0,0,0,66,1,0,0,0,0,0,0,72,42,0,0,0,0,66,1],[52,36,0,0,0,0,8,21,0,0,0,0,0,0,51,65,0,0,0,0,15,22,0,0,0,0,0,0,5,17,0,0,0,0,50,68],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,65,0,0,0,0,0,0,65],[13,4,0,0,0,0,67,60,0,0,0,0,0,0,0,0,9,60,0,0,0,0,63,64,0,0,8,0,0,0,0,0,0,8,0,0] >;
C4⋊C4.225D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{225}D_6 % in TeX
G:=Group("C4:C4.225D6"); // GroupNames label
G:=SmallGroup(192,523);
// by ID
G=gap.SmallGroup(192,523);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^-1*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations