metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.145D6, C6.932- (1+4), C6.742+ (1+4), Dic3⋊D4⋊43C2, D6⋊3Q8⋊34C2, (C2×D4).113D6, C4.4D4⋊17S3, (C2×Q8).108D6, C22⋊C4.38D6, C42⋊7S3⋊31C2, C23.9D6⋊47C2, C2.54(D4○D12), (C2×C6).228C24, D6⋊C4.73C22, C2.54(Q8○D12), C12.6Q8⋊29C2, C23.14D6⋊35C2, C2.78(D4⋊6D6), (C2×C12).633C23, (C4×C12).222C22, (C2×D12).35C22, (C6×D4).213C22, C4⋊Dic3.52C22, (C22×C6).58C23, C23.60(C22×S3), (C6×Q8).131C22, Dic3.D4⋊43C2, C22.D12⋊28C2, Dic3⋊C4.84C22, C22.249(S3×C23), (C2×Dic6).39C22, (C22×S3).100C23, C3⋊4(C22.56C24), (C2×Dic3).118C23, C6.D4.60C22, (C22×Dic3).147C22, (C3×C4.4D4)⋊20C2, (S3×C2×C4).123C22, (C2×C4).201(C22×S3), (C2×C3⋊D4).66C22, (C3×C22⋊C4).69C22, SmallGroup(192,1243)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 608 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×11], C22, C22 [×12], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23 [×2], C23 [×2], Dic3 [×6], C12 [×5], D6 [×6], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic6, C4×S3 [×2], D12, C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×3], C2×C12 [×2], C3×D4, C3×Q8, C22×S3 [×2], C22×C6 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C42.C2, Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×4], C6×D4, C6×Q8, C22.56C24, C12.6Q8, C42⋊7S3, Dic3.D4 [×2], C23.9D6 [×2], Dic3⋊D4 [×2], C22.D12 [×2], C23.14D6 [×2], D6⋊3Q8 [×2], C3×C4.4D4, C42.145D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×2], 2- (1+4), S3×C23, C22.56C24, D4⋊6D6, D4○D12, Q8○D12, C42.145D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 89 40 36)(2 31 41 96)(3 91 42 26)(4 33 43 86)(5 93 44 28)(6 35 45 88)(7 95 46 30)(8 25 47 90)(9 85 48 32)(10 27 37 92)(11 87 38 34)(12 29 39 94)(13 72 75 54)(14 49 76 67)(15 62 77 56)(16 51 78 69)(17 64 79 58)(18 53 80 71)(19 66 81 60)(20 55 82 61)(21 68 83 50)(22 57 84 63)(23 70 73 52)(24 59 74 65)
(1 16 7 22)(2 23 8 17)(3 18 9 24)(4 13 10 19)(5 20 11 14)(6 15 12 21)(25 64 31 70)(26 71 32 65)(27 66 33 72)(28 61 34 67)(29 68 35 62)(30 63 36 69)(37 81 43 75)(38 76 44 82)(39 83 45 77)(40 78 46 84)(41 73 47 79)(42 80 48 74)(49 93 55 87)(50 88 56 94)(51 95 57 89)(52 90 58 96)(53 85 59 91)(54 92 60 86)
(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 80 19 74)(14 73 20 79)(15 78 21 84)(16 83 22 77)(17 76 23 82)(18 81 24 75)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 42 43 48)(38 47 44 41)(39 40 45 46)(49 70 55 64)(50 63 56 69)(51 68 57 62)(52 61 58 67)(53 66 59 72)(54 71 60 65)(85 92 91 86)(87 90 93 96)(88 95 94 89)
G:=sub<Sym(96)| (1,89,40,36)(2,31,41,96)(3,91,42,26)(4,33,43,86)(5,93,44,28)(6,35,45,88)(7,95,46,30)(8,25,47,90)(9,85,48,32)(10,27,37,92)(11,87,38,34)(12,29,39,94)(13,72,75,54)(14,49,76,67)(15,62,77,56)(16,51,78,69)(17,64,79,58)(18,53,80,71)(19,66,81,60)(20,55,82,61)(21,68,83,50)(22,57,84,63)(23,70,73,52)(24,59,74,65), (1,16,7,22)(2,23,8,17)(3,18,9,24)(4,13,10,19)(5,20,11,14)(6,15,12,21)(25,64,31,70)(26,71,32,65)(27,66,33,72)(28,61,34,67)(29,68,35,62)(30,63,36,69)(37,81,43,75)(38,76,44,82)(39,83,45,77)(40,78,46,84)(41,73,47,79)(42,80,48,74)(49,93,55,87)(50,88,56,94)(51,95,57,89)(52,90,58,96)(53,85,59,91)(54,92,60,86), (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)(13,80,19,74)(14,73,20,79)(15,78,21,84)(16,83,22,77)(17,76,23,82)(18,81,24,75)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,70,55,64)(50,63,56,69)(51,68,57,62)(52,61,58,67)(53,66,59,72)(54,71,60,65)(85,92,91,86)(87,90,93,96)(88,95,94,89)>;
G:=Group( (1,89,40,36)(2,31,41,96)(3,91,42,26)(4,33,43,86)(5,93,44,28)(6,35,45,88)(7,95,46,30)(8,25,47,90)(9,85,48,32)(10,27,37,92)(11,87,38,34)(12,29,39,94)(13,72,75,54)(14,49,76,67)(15,62,77,56)(16,51,78,69)(17,64,79,58)(18,53,80,71)(19,66,81,60)(20,55,82,61)(21,68,83,50)(22,57,84,63)(23,70,73,52)(24,59,74,65), (1,16,7,22)(2,23,8,17)(3,18,9,24)(4,13,10,19)(5,20,11,14)(6,15,12,21)(25,64,31,70)(26,71,32,65)(27,66,33,72)(28,61,34,67)(29,68,35,62)(30,63,36,69)(37,81,43,75)(38,76,44,82)(39,83,45,77)(40,78,46,84)(41,73,47,79)(42,80,48,74)(49,93,55,87)(50,88,56,94)(51,95,57,89)(52,90,58,96)(53,85,59,91)(54,92,60,86), (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)(13,80,19,74)(14,73,20,79)(15,78,21,84)(16,83,22,77)(17,76,23,82)(18,81,24,75)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,70,55,64)(50,63,56,69)(51,68,57,62)(52,61,58,67)(53,66,59,72)(54,71,60,65)(85,92,91,86)(87,90,93,96)(88,95,94,89) );
G=PermutationGroup([(1,89,40,36),(2,31,41,96),(3,91,42,26),(4,33,43,86),(5,93,44,28),(6,35,45,88),(7,95,46,30),(8,25,47,90),(9,85,48,32),(10,27,37,92),(11,87,38,34),(12,29,39,94),(13,72,75,54),(14,49,76,67),(15,62,77,56),(16,51,78,69),(17,64,79,58),(18,53,80,71),(19,66,81,60),(20,55,82,61),(21,68,83,50),(22,57,84,63),(23,70,73,52),(24,59,74,65)], [(1,16,7,22),(2,23,8,17),(3,18,9,24),(4,13,10,19),(5,20,11,14),(6,15,12,21),(25,64,31,70),(26,71,32,65),(27,66,33,72),(28,61,34,67),(29,68,35,62),(30,63,36,69),(37,81,43,75),(38,76,44,82),(39,83,45,77),(40,78,46,84),(41,73,47,79),(42,80,48,74),(49,93,55,87),(50,88,56,94),(51,95,57,89),(52,90,58,96),(53,85,59,91),(54,92,60,86)], [(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),(13,80,19,74),(14,73,20,79),(15,78,21,84),(16,83,22,77),(17,76,23,82),(18,81,24,75),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,42,43,48),(38,47,44,41),(39,40,45,46),(49,70,55,64),(50,63,56,69),(51,68,57,62),(52,61,58,67),(53,66,59,72),(54,71,60,65),(85,92,91,86),(87,90,93,96),(88,95,94,89)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 6 | 6 | 9 | 9 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 7 | 10 | 7 | 0 | 0 | 0 | 0 |
| 10 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 11 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 11 |
| 8 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| 8 | 8 | 8 | 3 | 0 | 0 | 0 | 0 |
| 5 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 8 | 8 | 3 | 0 | 0 | 0 | 0 |
| 8 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [6,3,0,10,0,0,0,0,6,3,7,0,0,0,0,0,9,0,10,10,0,0,0,0,9,9,7,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0],[0,12,5,4,0,0,0,0,0,12,5,5,0,0,0,0,12,11,1,1,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,4,11],[8,8,5,5,0,0,0,0,8,8,0,5,0,0,0,0,3,8,5,5,0,0,0,0,8,3,5,5,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0],[8,8,5,0,0,0,0,0,8,8,0,5,0,0,0,0,8,3,5,5,0,0,0,0,3,8,5,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4E | 4F | ··· | 4K | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ (1+4) | 2- (1+4) | D4⋊6D6 | D4○D12 | Q8○D12 |
| kernel | C42.145D6 | C12.6Q8 | C42⋊7S3 | Dic3.D4 | C23.9D6 | Dic3⋊D4 | C22.D12 | C23.14D6 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{145}D_6 % in TeX
G:=Group("C4^2.145D6"); // GroupNames label
G:=SmallGroup(192,1243);
// by ID
G=gap.SmallGroup(192,1243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,297,136,6278]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations