metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.122D6, C6.72- 1+4, (C4×Q8)⋊9S3, (Q8×C12)⋊5C2, C4⋊C4.291D6, D6⋊Q8⋊10C2, (C4×Dic6)⋊36C2, (C2×Q8).200D6, C42⋊3S3⋊17C2, C42⋊2S3⋊33C2, Dic3⋊Q8⋊9C2, C4.18(C4○D12), (C2×C6).112C24, D6⋊C4.68C22, Dic6⋊C4⋊17C2, Dic3⋊5D4.10C2, C12.116(C4○D4), (C2×C12).621C23, C42⋊7S3.10C2, (C4×C12).238C22, C12.23D4.7C2, (C6×Q8).212C22, Dic3.21(C4○D4), (C2×D12).140C22, Dic3⋊C4.68C22, (C22×S3).44C23, C4⋊Dic3.303C22, C22.137(S3×C23), (C4×Dic3).81C22, C2.10(Q8.15D6), C3⋊3(C22.50C24), (C2×Dic6).147C22, (C2×Dic3).211C23, C4⋊C4⋊S3⋊10C2, C2.27(S3×C4○D4), C6.53(C2×C4○D4), C2.60(C2×C4○D12), (S3×C2×C4).206C22, (C3×C4⋊C4).340C22, (C2×C4).653(C22×S3), SmallGroup(192,1127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.122D6
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=a2b-1, dcd=c5 >
Subgroups: 488 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, C22.50C24, C4×Dic6, C42⋊2S3, C42⋊7S3, C42⋊3S3, Dic6⋊C4, Dic3⋊5D4, D6⋊Q8, C4⋊C4⋊S3, Dic3⋊Q8, C12.23D4, Q8×C12, C42.122D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.50C24, C2×C4○D12, Q8.15D6, S3×C4○D4, C42.122D6
►On 96 points
(1 57 17 86)(2 58 18 87)(3 59 19 88)(4 60 20 89)(5 49 21 90)(6 50 22 91)(7 51 23 92)(8 52 24 93)(9 53 13 94)(10 54 14 95)(11 55 15 96)(12 56 16 85)(25 61 44 78)(26 62 45 79)(27 63 46 80)(28 64 47 81)(29 65 48 82)(30 66 37 83)(31 67 38 84)(32 68 39 73)(33 69 40 74)(34 70 41 75)(35 71 42 76)(36 72 43 77)
(1 78 7 84)(2 68 8 62)(3 80 9 74)(4 70 10 64)(5 82 11 76)(6 72 12 66)(13 69 19 63)(14 81 20 75)(15 71 21 65)(16 83 22 77)(17 61 23 67)(18 73 24 79)(25 51 31 57)(26 87 32 93)(27 53 33 59)(28 89 34 95)(29 55 35 49)(30 91 36 85)(37 50 43 56)(38 86 44 92)(39 52 45 58)(40 88 46 94)(41 54 47 60)(42 90 48 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 77)(2 82)(3 75)(4 80)(5 73)(6 78)(7 83)(8 76)(9 81)(10 74)(11 79)(12 84)(13 64)(14 69)(15 62)(16 67)(17 72)(18 65)(19 70)(20 63)(21 68)(22 61)(23 66)(24 71)(25 56)(26 49)(27 54)(28 59)(29 52)(30 57)(31 50)(32 55)(33 60)(34 53)(35 58)(36 51)(37 86)(38 91)(39 96)(40 89)(41 94)(42 87)(43 92)(44 85)(45 90)(46 95)(47 88)(48 93)
G:=sub<Sym(96)| (1,57,17,86)(2,58,18,87)(3,59,19,88)(4,60,20,89)(5,49,21,90)(6,50,22,91)(7,51,23,92)(8,52,24,93)(9,53,13,94)(10,54,14,95)(11,55,15,96)(12,56,16,85)(25,61,44,78)(26,62,45,79)(27,63,46,80)(28,64,47,81)(29,65,48,82)(30,66,37,83)(31,67,38,84)(32,68,39,73)(33,69,40,74)(34,70,41,75)(35,71,42,76)(36,72,43,77), (1,78,7,84)(2,68,8,62)(3,80,9,74)(4,70,10,64)(5,82,11,76)(6,72,12,66)(13,69,19,63)(14,81,20,75)(15,71,21,65)(16,83,22,77)(17,61,23,67)(18,73,24,79)(25,51,31,57)(26,87,32,93)(27,53,33,59)(28,89,34,95)(29,55,35,49)(30,91,36,85)(37,50,43,56)(38,86,44,92)(39,52,45,58)(40,88,46,94)(41,54,47,60)(42,90,48,96), (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), (1,77)(2,82)(3,75)(4,80)(5,73)(6,78)(7,83)(8,76)(9,81)(10,74)(11,79)(12,84)(13,64)(14,69)(15,62)(16,67)(17,72)(18,65)(19,70)(20,63)(21,68)(22,61)(23,66)(24,71)(25,56)(26,49)(27,54)(28,59)(29,52)(30,57)(31,50)(32,55)(33,60)(34,53)(35,58)(36,51)(37,86)(38,91)(39,96)(40,89)(41,94)(42,87)(43,92)(44,85)(45,90)(46,95)(47,88)(48,93)>;
G:=Group( (1,57,17,86)(2,58,18,87)(3,59,19,88)(4,60,20,89)(5,49,21,90)(6,50,22,91)(7,51,23,92)(8,52,24,93)(9,53,13,94)(10,54,14,95)(11,55,15,96)(12,56,16,85)(25,61,44,78)(26,62,45,79)(27,63,46,80)(28,64,47,81)(29,65,48,82)(30,66,37,83)(31,67,38,84)(32,68,39,73)(33,69,40,74)(34,70,41,75)(35,71,42,76)(36,72,43,77), (1,78,7,84)(2,68,8,62)(3,80,9,74)(4,70,10,64)(5,82,11,76)(6,72,12,66)(13,69,19,63)(14,81,20,75)(15,71,21,65)(16,83,22,77)(17,61,23,67)(18,73,24,79)(25,51,31,57)(26,87,32,93)(27,53,33,59)(28,89,34,95)(29,55,35,49)(30,91,36,85)(37,50,43,56)(38,86,44,92)(39,52,45,58)(40,88,46,94)(41,54,47,60)(42,90,48,96), (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), (1,77)(2,82)(3,75)(4,80)(5,73)(6,78)(7,83)(8,76)(9,81)(10,74)(11,79)(12,84)(13,64)(14,69)(15,62)(16,67)(17,72)(18,65)(19,70)(20,63)(21,68)(22,61)(23,66)(24,71)(25,56)(26,49)(27,54)(28,59)(29,52)(30,57)(31,50)(32,55)(33,60)(34,53)(35,58)(36,51)(37,86)(38,91)(39,96)(40,89)(41,94)(42,87)(43,92)(44,85)(45,90)(46,95)(47,88)(48,93) );
G=PermutationGroup([[(1,57,17,86),(2,58,18,87),(3,59,19,88),(4,60,20,89),(5,49,21,90),(6,50,22,91),(7,51,23,92),(8,52,24,93),(9,53,13,94),(10,54,14,95),(11,55,15,96),(12,56,16,85),(25,61,44,78),(26,62,45,79),(27,63,46,80),(28,64,47,81),(29,65,48,82),(30,66,37,83),(31,67,38,84),(32,68,39,73),(33,69,40,74),(34,70,41,75),(35,71,42,76),(36,72,43,77)], [(1,78,7,84),(2,68,8,62),(3,80,9,74),(4,70,10,64),(5,82,11,76),(6,72,12,66),(13,69,19,63),(14,81,20,75),(15,71,21,65),(16,83,22,77),(17,61,23,67),(18,73,24,79),(25,51,31,57),(26,87,32,93),(27,53,33,59),(28,89,34,95),(29,55,35,49),(30,91,36,85),(37,50,43,56),(38,86,44,92),(39,52,45,58),(40,88,46,94),(41,54,47,60),(42,90,48,96)], [(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)], [(1,77),(2,82),(3,75),(4,80),(5,73),(6,78),(7,83),(8,76),(9,81),(10,74),(11,79),(12,84),(13,64),(14,69),(15,62),(16,67),(17,72),(18,65),(19,70),(20,63),(21,68),(22,61),(23,66),(24,71),(25,56),(26,49),(27,54),(28,59),(29,52),(30,57),(31,50),(32,55),(33,60),(34,53),(35,58),(36,51),(37,86),(38,91),(39,96),(40,89),(41,94),(42,87),(43,92),(44,85),(45,90),(46,95),(47,88),(48,93)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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○D4 | C4○D12 | 2- 1+4 | Q8.15D6 | S3×C4○D4 |
| kernel | C42.122D6 | C4×Dic6 | C42⋊2S3 | C42⋊7S3 | C42⋊3S3 | Dic6⋊C4 | Dic3⋊5D4 | D6⋊Q8 | C4⋊C4⋊S3 | Dic3⋊Q8 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.122D6 ►in GL4(𝔽13) generated by
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
| 8 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [3,7,0,0,6,10,0,0,0,0,5,0,0,0,0,5],[8,0,0,0,0,8,0,0,0,0,12,8,0,0,0,1],[8,5,0,0,8,0,0,0,0,0,12,0,0,0,3,1],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,10,12] >;
C42.122D6 in GAP, Magma, Sage, TeX
C_4^2._{122}D_6 % in TeX
G:=Group("C4^2.122D6"); // GroupNames label
G:=SmallGroup(192,1127);
// by ID
G=gap.SmallGroup(192,1127);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,232,758,100,794,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^5>;
// generators/relations