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, C42⋊2S3.9C2, (C2×C6).275C24, D6⋊3Q8.13C2, C12.6Q8⋊25C2, C12.138(C4○D4), C4.42(D4⋊2S3), (C4×C12).216C22, (C2×C12).108C23, 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), (C2×Dic3).273C23, (C4×Dic3).164C22, (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
Subgroups: 400 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×13], C22, C22 [×3], S3, C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, Dic3 [×7], C12 [×2], C12 [×6], D6 [×3], C2×C6, C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C2×Q8 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4 [×10], C4⋊Dic3 [×6], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4, C6×Q8 [×2], C22.35C24, C12.6Q8, C42⋊2S3, Dic3.Q8 [×4], C4⋊C4⋊S3 [×4], Q8×Dic3 [×2], D6⋊3Q8 [×2], C3×C4⋊Q8, C42.176D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2- (1+4) [×2], D4⋊2S3 [×2], S3×C23, C22.35C24, C2×D4⋊2S3, Q8.15D6 [×2], C42.176D6
Generators and relations
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 >
►On 96 points
(1 30 7 36)(2 25 8 31)(3 32 9 26)(4 27 10 33)(5 34 11 28)(6 29 12 35)(13 41 19 47)(14 48 20 42)(15 43 21 37)(16 38 22 44)(17 45 23 39)(18 40 24 46)(49 64 55 70)(50 71 56 65)(51 66 57 72)(52 61 58 67)(53 68 59 62)(54 63 60 69)(73 88 79 94)(74 95 80 89)(75 90 81 96)(76 85 82 91)(77 92 83 86)(78 87 84 93)
(1 19 53 94)(2 95 54 20)(3 21 55 96)(4 85 56 22)(5 23 57 86)(6 87 58 24)(7 13 59 88)(8 89 60 14)(9 15 49 90)(10 91 50 16)(11 17 51 92)(12 93 52 18)(25 80 63 42)(26 43 64 81)(27 82 65 44)(28 45 66 83)(29 84 67 46)(30 47 68 73)(31 74 69 48)(32 37 70 75)(33 76 71 38)(34 39 72 77)(35 78 61 40)(36 41 62 79)
(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 59 52)(2 51 60 5)(3 4 49 50)(7 12 53 58)(8 57 54 11)(9 10 55 56)(13 24 94 93)(14 92 95 23)(15 22 96 91)(16 90 85 21)(17 20 86 89)(18 88 87 19)(25 72 69 28)(26 27 70 71)(29 68 61 36)(30 35 62 67)(31 66 63 34)(32 33 64 65)(37 44 81 76)(38 75 82 43)(39 42 83 74)(40 73 84 41)(45 48 77 80)(46 79 78 47)
G:=sub<Sym(96)| (1,30,7,36)(2,25,8,31)(3,32,9,26)(4,27,10,33)(5,34,11,28)(6,29,12,35)(13,41,19,47)(14,48,20,42)(15,43,21,37)(16,38,22,44)(17,45,23,39)(18,40,24,46)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69)(73,88,79,94)(74,95,80,89)(75,90,81,96)(76,85,82,91)(77,92,83,86)(78,87,84,93), (1,19,53,94)(2,95,54,20)(3,21,55,96)(4,85,56,22)(5,23,57,86)(6,87,58,24)(7,13,59,88)(8,89,60,14)(9,15,49,90)(10,91,50,16)(11,17,51,92)(12,93,52,18)(25,80,63,42)(26,43,64,81)(27,82,65,44)(28,45,66,83)(29,84,67,46)(30,47,68,73)(31,74,69,48)(32,37,70,75)(33,76,71,38)(34,39,72,77)(35,78,61,40)(36,41,62,79), (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), (1,6,59,52)(2,51,60,5)(3,4,49,50)(7,12,53,58)(8,57,54,11)(9,10,55,56)(13,24,94,93)(14,92,95,23)(15,22,96,91)(16,90,85,21)(17,20,86,89)(18,88,87,19)(25,72,69,28)(26,27,70,71)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(37,44,81,76)(38,75,82,43)(39,42,83,74)(40,73,84,41)(45,48,77,80)(46,79,78,47)>;
G:=Group( (1,30,7,36)(2,25,8,31)(3,32,9,26)(4,27,10,33)(5,34,11,28)(6,29,12,35)(13,41,19,47)(14,48,20,42)(15,43,21,37)(16,38,22,44)(17,45,23,39)(18,40,24,46)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69)(73,88,79,94)(74,95,80,89)(75,90,81,96)(76,85,82,91)(77,92,83,86)(78,87,84,93), (1,19,53,94)(2,95,54,20)(3,21,55,96)(4,85,56,22)(5,23,57,86)(6,87,58,24)(7,13,59,88)(8,89,60,14)(9,15,49,90)(10,91,50,16)(11,17,51,92)(12,93,52,18)(25,80,63,42)(26,43,64,81)(27,82,65,44)(28,45,66,83)(29,84,67,46)(30,47,68,73)(31,74,69,48)(32,37,70,75)(33,76,71,38)(34,39,72,77)(35,78,61,40)(36,41,62,79), (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), (1,6,59,52)(2,51,60,5)(3,4,49,50)(7,12,53,58)(8,57,54,11)(9,10,55,56)(13,24,94,93)(14,92,95,23)(15,22,96,91)(16,90,85,21)(17,20,86,89)(18,88,87,19)(25,72,69,28)(26,27,70,71)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(37,44,81,76)(38,75,82,43)(39,42,83,74)(40,73,84,41)(45,48,77,80)(46,79,78,47) );
G=PermutationGroup([(1,30,7,36),(2,25,8,31),(3,32,9,26),(4,27,10,33),(5,34,11,28),(6,29,12,35),(13,41,19,47),(14,48,20,42),(15,43,21,37),(16,38,22,44),(17,45,23,39),(18,40,24,46),(49,64,55,70),(50,71,56,65),(51,66,57,72),(52,61,58,67),(53,68,59,62),(54,63,60,69),(73,88,79,94),(74,95,80,89),(75,90,81,96),(76,85,82,91),(77,92,83,86),(78,87,84,93)], [(1,19,53,94),(2,95,54,20),(3,21,55,96),(4,85,56,22),(5,23,57,86),(6,87,58,24),(7,13,59,88),(8,89,60,14),(9,15,49,90),(10,91,50,16),(11,17,51,92),(12,93,52,18),(25,80,63,42),(26,43,64,81),(27,82,65,44),(28,45,66,83),(29,84,67,46),(30,47,68,73),(31,74,69,48),(32,37,70,75),(33,76,71,38),(34,39,72,77),(35,78,61,40),(36,41,62,79)], [(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)], [(1,6,59,52),(2,51,60,5),(3,4,49,50),(7,12,53,58),(8,57,54,11),(9,10,55,56),(13,24,94,93),(14,92,95,23),(15,22,96,91),(16,90,85,21),(17,20,86,89),(18,88,87,19),(25,72,69,28),(26,27,70,71),(29,68,61,36),(30,35,62,67),(31,66,63,34),(32,33,64,65),(37,44,81,76),(38,75,82,43),(39,42,83,74),(40,73,84,41),(45,48,77,80),(46,79,78,47)])
Matrix representation ►G ⊆ 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] >;
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 |
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