metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.148D6, C6.952- (1+4), C6.1302+ (1+4), (C4×S3)⋊1Q8, C12⋊Q8⋊35C2, D6.4(C2×Q8), C4.39(S3×Q8), C4⋊C4.111D6, C42.C2⋊4S3, C12.50(C2×Q8), C12⋊2Q8⋊32C2, D6⋊Q8.2C2, Dic3.6(C2×Q8), C2.55(D4○D12), C6.42(C22×Q8), Dic3.Q8⋊32C2, (C2×C6).233C24, C42⋊2S3.6C2, (C2×C12).87C23, D6⋊C4.39C22, C4.D12.11C2, C2.57(Q8○D12), C12.3Q8⋊33C2, (C4×C12).193C22, C4⋊Dic3.240C22, C22.254(S3×C23), (C2×Dic6).40C22, Dic3⋊C4.122C22, (C22×S3).220C23, C3⋊4(C23.41C23), (C4×Dic3).140C22, (C2×Dic3).121C23, C2.25(C2×S3×Q8), (S3×C4⋊C4).11C2, C4⋊C4⋊7S3.12C2, (C3×C42.C2)⋊6C2, (S3×C2×C4).124C22, (C3×C4⋊C4).188C22, (C2×C4).203(C22×S3), SmallGroup(192,1248)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×14], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×6], D6 [×2], D6 [×2], C2×C6, C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C22×C4 [×3], C2×Q8 [×4], Dic6 [×4], C4×S3 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C22×S3, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2, C42.C2 [×3], C4⋊Q8 [×4], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×4], D6⋊C4 [×2], D6⋊C4 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6 [×4], S3×C2×C4, S3×C2×C4 [×2], C23.41C23, C12⋊2Q8, C42⋊2S3, C12⋊Q8, C12⋊Q8 [×2], Dic3.Q8 [×2], C12.3Q8, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8 [×2], C4.D12 [×2], C3×C42.C2, C42.148D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, C22×S3 [×7], C22×Q8, 2+ (1+4), 2- (1+4), S3×Q8 [×2], S3×C23, C23.41C23, C2×S3×Q8, D4○D12, Q8○D12, C42.148D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 81 45 32)(2 33 46 82)(3 83 47 34)(4 35 48 84)(5 73 37 36)(6 25 38 74)(7 75 39 26)(8 27 40 76)(9 77 41 28)(10 29 42 78)(11 79 43 30)(12 31 44 80)(13 65 50 92)(14 93 51 66)(15 67 52 94)(16 95 53 68)(17 69 54 96)(18 85 55 70)(19 71 56 86)(20 87 57 72)(21 61 58 88)(22 89 59 62)(23 63 60 90)(24 91 49 64)
(1 71 7 65)(2 93 8 87)(3 61 9 67)(4 95 10 89)(5 63 11 69)(6 85 12 91)(13 32 19 26)(14 76 20 82)(15 34 21 28)(16 78 22 84)(17 36 23 30)(18 80 24 74)(25 55 31 49)(27 57 33 51)(29 59 35 53)(37 90 43 96)(38 70 44 64)(39 92 45 86)(40 72 46 66)(41 94 47 88)(42 62 48 68)(50 81 56 75)(52 83 58 77)(54 73 60 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 7 12)(2 11 8 5)(3 4 9 10)(13 55 19 49)(14 60 20 54)(15 53 21 59)(16 58 22 52)(17 51 23 57)(18 56 24 50)(25 26 31 32)(27 36 33 30)(28 29 34 35)(37 46 43 40)(38 39 44 45)(41 42 47 48)(61 89 67 95)(62 94 68 88)(63 87 69 93)(64 92 70 86)(65 85 71 91)(66 90 72 96)(73 82 79 76)(74 75 80 81)(77 78 83 84)
G:=sub<Sym(96)| (1,81,45,32)(2,33,46,82)(3,83,47,34)(4,35,48,84)(5,73,37,36)(6,25,38,74)(7,75,39,26)(8,27,40,76)(9,77,41,28)(10,29,42,78)(11,79,43,30)(12,31,44,80)(13,65,50,92)(14,93,51,66)(15,67,52,94)(16,95,53,68)(17,69,54,96)(18,85,55,70)(19,71,56,86)(20,87,57,72)(21,61,58,88)(22,89,59,62)(23,63,60,90)(24,91,49,64), (1,71,7,65)(2,93,8,87)(3,61,9,67)(4,95,10,89)(5,63,11,69)(6,85,12,91)(13,32,19,26)(14,76,20,82)(15,34,21,28)(16,78,22,84)(17,36,23,30)(18,80,24,74)(25,55,31,49)(27,57,33,51)(29,59,35,53)(37,90,43,96)(38,70,44,64)(39,92,45,86)(40,72,46,66)(41,94,47,88)(42,62,48,68)(50,81,56,75)(52,83,58,77)(54,73,60,79), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,89,67,95)(62,94,68,88)(63,87,69,93)(64,92,70,86)(65,85,71,91)(66,90,72,96)(73,82,79,76)(74,75,80,81)(77,78,83,84)>;
G:=Group( (1,81,45,32)(2,33,46,82)(3,83,47,34)(4,35,48,84)(5,73,37,36)(6,25,38,74)(7,75,39,26)(8,27,40,76)(9,77,41,28)(10,29,42,78)(11,79,43,30)(12,31,44,80)(13,65,50,92)(14,93,51,66)(15,67,52,94)(16,95,53,68)(17,69,54,96)(18,85,55,70)(19,71,56,86)(20,87,57,72)(21,61,58,88)(22,89,59,62)(23,63,60,90)(24,91,49,64), (1,71,7,65)(2,93,8,87)(3,61,9,67)(4,95,10,89)(5,63,11,69)(6,85,12,91)(13,32,19,26)(14,76,20,82)(15,34,21,28)(16,78,22,84)(17,36,23,30)(18,80,24,74)(25,55,31,49)(27,57,33,51)(29,59,35,53)(37,90,43,96)(38,70,44,64)(39,92,45,86)(40,72,46,66)(41,94,47,88)(42,62,48,68)(50,81,56,75)(52,83,58,77)(54,73,60,79), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,89,67,95)(62,94,68,88)(63,87,69,93)(64,92,70,86)(65,85,71,91)(66,90,72,96)(73,82,79,76)(74,75,80,81)(77,78,83,84) );
G=PermutationGroup([(1,81,45,32),(2,33,46,82),(3,83,47,34),(4,35,48,84),(5,73,37,36),(6,25,38,74),(7,75,39,26),(8,27,40,76),(9,77,41,28),(10,29,42,78),(11,79,43,30),(12,31,44,80),(13,65,50,92),(14,93,51,66),(15,67,52,94),(16,95,53,68),(17,69,54,96),(18,85,55,70),(19,71,56,86),(20,87,57,72),(21,61,58,88),(22,89,59,62),(23,63,60,90),(24,91,49,64)], [(1,71,7,65),(2,93,8,87),(3,61,9,67),(4,95,10,89),(5,63,11,69),(6,85,12,91),(13,32,19,26),(14,76,20,82),(15,34,21,28),(16,78,22,84),(17,36,23,30),(18,80,24,74),(25,55,31,49),(27,57,33,51),(29,59,35,53),(37,90,43,96),(38,70,44,64),(39,92,45,86),(40,72,46,66),(41,94,47,88),(42,62,48,68),(50,81,56,75),(52,83,58,77),(54,73,60,79)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,55,19,49),(14,60,20,54),(15,53,21,59),(16,58,22,52),(17,51,23,57),(18,56,24,50),(25,26,31,32),(27,36,33,30),(28,29,34,35),(37,46,43,40),(38,39,44,45),(41,42,47,48),(61,89,67,95),(62,94,68,88),(63,87,69,93),(64,92,70,86),(65,85,71,91),(66,90,72,96),(73,82,79,76),(74,75,80,81),(77,78,83,84)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,8,0,0,0,0,0,0,8,0,0],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,12,1,0,0,0,0,12,0,0,0],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,1,0,0,0,0,0,1,12,0,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | 2+ (1+4) | 2- (1+4) | S3×Q8 | D4○D12 | Q8○D12 |
| kernel | C42.148D6 | C12⋊2Q8 | C42⋊2S3 | C12⋊Q8 | Dic3.Q8 | C12.3Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6⋊Q8 | C4.D12 | C3×C42.C2 | C42.C2 | C4×S3 | C42 | C4⋊C4 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 6 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{148}D_6 % in TeX
G:=Group("C4^2.148D6"); // GroupNames label
G:=SmallGroup(192,1248);
// by ID
G=gap.SmallGroup(192,1248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations