metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.176D6, C6.382- 1+4, C4⋊Q8⋊14S3, C4⋊C4.125D6, (Q8×Dic3)⋊23C2, (C2×Q8).172D6, Dic3.Q8⋊42C2, (C2×C6).275C24, C42⋊2S3.9C2, D6⋊3Q8.13C2, C12.6Q8⋊25C2, C12.138(C4○D4), C4.42(D4⋊2S3), (C2×C12).108C23, (C4×C12).216C22, D6⋊C4.154C22, (C6×Q8).142C22, Dic3⋊C4.63C22, C4⋊Dic3.254C22, C22.296(S3×C23), (C22×S3).120C23, C2.39(Q8.15D6), C3⋊7(C22.35C24), (C4×Dic3).164C22, (C2×Dic3).273C23, (C3×C4⋊Q8)⋊17C2, C4⋊C4⋊S3.4C2, C6.101(C2×C4○D4), C2.65(C2×D4⋊2S3), (S3×C2×C4).148C22, (C3×C4⋊C4).218C22, (C2×C4).221(C22×S3), SmallGroup(192,1290)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.176D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c5 >
Subgroups: 400 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C6×Q8, C22.35C24, C12.6Q8, C42⋊2S3, Dic3.Q8, C4⋊C4⋊S3, Q8×Dic3, D6⋊3Q8, C3×C4⋊Q8, C42.176D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D4⋊2S3, S3×C23, C22.35C24, C2×D4⋊2S3, Q8.15D6, C42.176D6
►On 96 points
Generators in S96
(1 52 7 58)(2 59 8 53)(3 54 9 60)(4 49 10 55)(5 56 11 50)(6 51 12 57)(13 73 19 79)(14 80 20 74)(15 75 21 81)(16 82 22 76)(17 77 23 83)(18 84 24 78)(25 62 31 68)(26 69 32 63)(27 64 33 70)(28 71 34 65)(29 66 35 72)(30 61 36 67)(37 93 43 87)(38 88 44 94)(39 95 45 89)(40 90 46 96)(41 85 47 91)(42 92 48 86)
(1 64 80 41)(2 42 81 65)(3 66 82 43)(4 44 83 67)(5 68 84 45)(6 46 73 69)(7 70 74 47)(8 48 75 71)(9 72 76 37)(10 38 77 61)(11 62 78 39)(12 40 79 63)(13 26 57 90)(14 91 58 27)(15 28 59 92)(16 93 60 29)(17 30 49 94)(18 95 50 31)(19 32 51 96)(20 85 52 33)(21 34 53 86)(22 87 54 35)(23 36 55 88)(24 89 56 25)
(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 74 79)(2 78 75 5)(3 4 76 77)(7 12 80 73)(8 84 81 11)(9 10 82 83)(13 58 51 20)(14 19 52 57)(15 56 53 18)(16 17 54 55)(21 50 59 24)(22 23 60 49)(25 28 95 86)(26 85 96 27)(29 36 87 94)(30 93 88 35)(31 34 89 92)(32 91 90 33)(37 44 66 61)(38 72 67 43)(39 42 68 71)(40 70 69 41)(45 48 62 65)(46 64 63 47)
G:=sub<Sym(96)| (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,73,19,79)(14,80,20,74)(15,75,21,81)(16,82,22,76)(17,77,23,83)(18,84,24,78)(25,62,31,68)(26,69,32,63)(27,64,33,70)(28,71,34,65)(29,66,35,72)(30,61,36,67)(37,93,43,87)(38,88,44,94)(39,95,45,89)(40,90,46,96)(41,85,47,91)(42,92,48,86), (1,64,80,41)(2,42,81,65)(3,66,82,43)(4,44,83,67)(5,68,84,45)(6,46,73,69)(7,70,74,47)(8,48,75,71)(9,72,76,37)(10,38,77,61)(11,62,78,39)(12,40,79,63)(13,26,57,90)(14,91,58,27)(15,28,59,92)(16,93,60,29)(17,30,49,94)(18,95,50,31)(19,32,51,96)(20,85,52,33)(21,34,53,86)(22,87,54,35)(23,36,55,88)(24,89,56,25), (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), (1,6,74,79)(2,78,75,5)(3,4,76,77)(7,12,80,73)(8,84,81,11)(9,10,82,83)(13,58,51,20)(14,19,52,57)(15,56,53,18)(16,17,54,55)(21,50,59,24)(22,23,60,49)(25,28,95,86)(26,85,96,27)(29,36,87,94)(30,93,88,35)(31,34,89,92)(32,91,90,33)(37,44,66,61)(38,72,67,43)(39,42,68,71)(40,70,69,41)(45,48,62,65)(46,64,63,47)>;
G:=Group( (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,73,19,79)(14,80,20,74)(15,75,21,81)(16,82,22,76)(17,77,23,83)(18,84,24,78)(25,62,31,68)(26,69,32,63)(27,64,33,70)(28,71,34,65)(29,66,35,72)(30,61,36,67)(37,93,43,87)(38,88,44,94)(39,95,45,89)(40,90,46,96)(41,85,47,91)(42,92,48,86), (1,64,80,41)(2,42,81,65)(3,66,82,43)(4,44,83,67)(5,68,84,45)(6,46,73,69)(7,70,74,47)(8,48,75,71)(9,72,76,37)(10,38,77,61)(11,62,78,39)(12,40,79,63)(13,26,57,90)(14,91,58,27)(15,28,59,92)(16,93,60,29)(17,30,49,94)(18,95,50,31)(19,32,51,96)(20,85,52,33)(21,34,53,86)(22,87,54,35)(23,36,55,88)(24,89,56,25), (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), (1,6,74,79)(2,78,75,5)(3,4,76,77)(7,12,80,73)(8,84,81,11)(9,10,82,83)(13,58,51,20)(14,19,52,57)(15,56,53,18)(16,17,54,55)(21,50,59,24)(22,23,60,49)(25,28,95,86)(26,85,96,27)(29,36,87,94)(30,93,88,35)(31,34,89,92)(32,91,90,33)(37,44,66,61)(38,72,67,43)(39,42,68,71)(40,70,69,41)(45,48,62,65)(46,64,63,47) );
G=PermutationGroup([[(1,52,7,58),(2,59,8,53),(3,54,9,60),(4,49,10,55),(5,56,11,50),(6,51,12,57),(13,73,19,79),(14,80,20,74),(15,75,21,81),(16,82,22,76),(17,77,23,83),(18,84,24,78),(25,62,31,68),(26,69,32,63),(27,64,33,70),(28,71,34,65),(29,66,35,72),(30,61,36,67),(37,93,43,87),(38,88,44,94),(39,95,45,89),(40,90,46,96),(41,85,47,91),(42,92,48,86)], [(1,64,80,41),(2,42,81,65),(3,66,82,43),(4,44,83,67),(5,68,84,45),(6,46,73,69),(7,70,74,47),(8,48,75,71),(9,72,76,37),(10,38,77,61),(11,62,78,39),(12,40,79,63),(13,26,57,90),(14,91,58,27),(15,28,59,92),(16,93,60,29),(17,30,49,94),(18,95,50,31),(19,32,51,96),(20,85,52,33),(21,34,53,86),(22,87,54,35),(23,36,55,88),(24,89,56,25)], [(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)], [(1,6,74,79),(2,78,75,5),(3,4,76,77),(7,12,80,73),(8,84,81,11),(9,10,82,83),(13,58,51,20),(14,19,52,57),(15,56,53,18),(16,17,54,55),(21,50,59,24),(22,23,60,49),(25,28,95,86),(26,85,96,27),(29,36,87,94),(30,93,88,35),(31,34,89,92),(32,91,90,33),(37,44,66,61),(38,72,67,43),(39,42,68,71),(40,70,69,41),(45,48,62,65),(46,64,63,47)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2- 1+4 | D4⋊2S3 | Q8.15D6 |
| kernel | C42.176D6 | C12.6Q8 | C42⋊2S3 | Dic3.Q8 | C4⋊C4⋊S3 | Q8×Dic3 | D6⋊3Q8 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C42.176D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 4 |
| 0 | 0 | 7 | 7 | 9 | 4 |
| 0 | 0 | 0 | 4 | 12 | 7 |
| 0 | 0 | 9 | 4 | 6 | 6 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 4 | 9 |
| 0 | 0 | 12 | 7 | 0 | 9 |
| 0 | 0 | 9 | 4 | 6 | 6 |
| 0 | 0 | 0 | 4 | 12 | 7 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,2,4,0,0,0,0,9,11,0,0],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,7,0,9,0,0,6,7,4,4,0,0,0,9,12,6,0,0,4,4,7,6],[12,12,0,0,0,0,2,1,0,0,0,0,0,0,6,12,9,0,0,0,6,7,4,4,0,0,4,0,6,12,0,0,9,9,6,7] >;
C42.176D6 in GAP, Magma, Sage, TeX
C_4^2._{176}D_6 % in TeX
G:=Group("C4^2.176D6"); // GroupNames label
G:=SmallGroup(192,1290);
// by ID
G=gap.SmallGroup(192,1290);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=b^2*c^5>;
// generators/relations