metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.161D6, C6.1392+ (1+4), C6.1002- (1+4), (C4×D12)⋊15C2, C4⋊C4.118D6, C42⋊2C2⋊4S3, D6⋊Q8⋊43C2, C4.D12⋊41C2, Dic3⋊D4.4C2, C22⋊C4.79D6, C12.6Q8⋊9C2, D6.14(C4○D4), D6.D4⋊41C2, C23.9D6⋊51C2, C2.64(D4○D12), (C4×C12).33C22, (C2×C6).251C24, C2.64(Q8○D12), C12.3Q8⋊39C2, Dic3⋊4D4⋊36C2, (C2×C12).194C23, D6⋊C4.114C22, C4⋊Dic3.54C22, C23.67(C22×S3), (C22×C6).65C23, Dic3.D4⋊45C2, (C2×D12).227C22, C22.D12⋊29C2, Dic3⋊C4.56C22, C22.272(S3×C23), (C22×S3).225C23, C3⋊9(C22.33C24), (C2×Dic6).184C22, (C2×Dic3).265C23, (C4×Dic3).151C22, C6.D4.67C22, (C22×Dic3).151C22, (S3×C4⋊C4)⋊41C2, C4⋊C4⋊S3⋊42C2, C2.98(S3×C4○D4), C6.209(C2×C4○D4), (C3×C42⋊2C2)⋊6C2, (S3×C2×C4).219C22, (C3×C4⋊C4).203C22, (C2×C4).209(C22×S3), (C2×C3⋊D4).71C22, (C3×C22⋊C4).76C22, SmallGroup(192,1266)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 560 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], Dic3 [×6], C12 [×6], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C42, C42, C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×11], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic6, C4×S3 [×5], D12 [×2], C2×Dic3 [×6], C2×Dic3, C3⋊D4 [×3], C2×C12 [×6], C22×S3 [×2], C22×C6, C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3 [×5], D6⋊C4 [×6], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×4], C2×D12, C22×Dic3, C2×C3⋊D4 [×2], C22.33C24, C12.6Q8, C4×D12, Dic3.D4, Dic3⋊4D4, C23.9D6 [×2], Dic3⋊D4, C22.D12, C12.3Q8, S3×C4⋊C4, D6.D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.161D6
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- (1+4), S3×C23, C22.33C24, S3×C4○D4, D4○D12, Q8○D12, C42.161D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 88 23 54)(2 95 24 49)(3 90 13 56)(4 85 14 51)(5 92 15 58)(6 87 16 53)(7 94 17 60)(8 89 18 55)(9 96 19 50)(10 91 20 57)(11 86 21 52)(12 93 22 59)(25 74 67 41)(26 81 68 48)(27 76 69 43)(28 83 70 38)(29 78 71 45)(30 73 72 40)(31 80 61 47)(32 75 62 42)(33 82 63 37)(34 77 64 44)(35 84 65 39)(36 79 66 46)
(1 62 7 68)(2 27 8 33)(3 64 9 70)(4 29 10 35)(5 66 11 72)(6 31 12 25)(13 34 19 28)(14 71 20 65)(15 36 21 30)(16 61 22 67)(17 26 23 32)(18 63 24 69)(37 49 43 55)(38 90 44 96)(39 51 45 57)(40 92 46 86)(41 53 47 59)(42 94 48 88)(50 83 56 77)(52 73 58 79)(54 75 60 81)(74 87 80 93)(76 89 82 95)(78 91 84 85)
(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 61 7 67)(2 66 8 72)(3 71 9 65)(4 64 10 70)(5 69 11 63)(6 62 12 68)(13 29 19 35)(14 34 20 28)(15 27 21 33)(16 32 22 26)(17 25 23 31)(18 30 24 36)(37 86 43 92)(38 91 44 85)(39 96 45 90)(40 89 46 95)(41 94 47 88)(42 87 48 93)(49 73 55 79)(50 78 56 84)(51 83 57 77)(52 76 58 82)(53 81 59 75)(54 74 60 80)
G:=sub<Sym(96)| (1,88,23,54)(2,95,24,49)(3,90,13,56)(4,85,14,51)(5,92,15,58)(6,87,16,53)(7,94,17,60)(8,89,18,55)(9,96,19,50)(10,91,20,57)(11,86,21,52)(12,93,22,59)(25,74,67,41)(26,81,68,48)(27,76,69,43)(28,83,70,38)(29,78,71,45)(30,73,72,40)(31,80,61,47)(32,75,62,42)(33,82,63,37)(34,77,64,44)(35,84,65,39)(36,79,66,46), (1,62,7,68)(2,27,8,33)(3,64,9,70)(4,29,10,35)(5,66,11,72)(6,31,12,25)(13,34,19,28)(14,71,20,65)(15,36,21,30)(16,61,22,67)(17,26,23,32)(18,63,24,69)(37,49,43,55)(38,90,44,96)(39,51,45,57)(40,92,46,86)(41,53,47,59)(42,94,48,88)(50,83,56,77)(52,73,58,79)(54,75,60,81)(74,87,80,93)(76,89,82,95)(78,91,84,85), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,29,19,35)(14,34,20,28)(15,27,21,33)(16,32,22,26)(17,25,23,31)(18,30,24,36)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93)(49,73,55,79)(50,78,56,84)(51,83,57,77)(52,76,58,82)(53,81,59,75)(54,74,60,80)>;
G:=Group( (1,88,23,54)(2,95,24,49)(3,90,13,56)(4,85,14,51)(5,92,15,58)(6,87,16,53)(7,94,17,60)(8,89,18,55)(9,96,19,50)(10,91,20,57)(11,86,21,52)(12,93,22,59)(25,74,67,41)(26,81,68,48)(27,76,69,43)(28,83,70,38)(29,78,71,45)(30,73,72,40)(31,80,61,47)(32,75,62,42)(33,82,63,37)(34,77,64,44)(35,84,65,39)(36,79,66,46), (1,62,7,68)(2,27,8,33)(3,64,9,70)(4,29,10,35)(5,66,11,72)(6,31,12,25)(13,34,19,28)(14,71,20,65)(15,36,21,30)(16,61,22,67)(17,26,23,32)(18,63,24,69)(37,49,43,55)(38,90,44,96)(39,51,45,57)(40,92,46,86)(41,53,47,59)(42,94,48,88)(50,83,56,77)(52,73,58,79)(54,75,60,81)(74,87,80,93)(76,89,82,95)(78,91,84,85), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,29,19,35)(14,34,20,28)(15,27,21,33)(16,32,22,26)(17,25,23,31)(18,30,24,36)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93)(49,73,55,79)(50,78,56,84)(51,83,57,77)(52,76,58,82)(53,81,59,75)(54,74,60,80) );
G=PermutationGroup([(1,88,23,54),(2,95,24,49),(3,90,13,56),(4,85,14,51),(5,92,15,58),(6,87,16,53),(7,94,17,60),(8,89,18,55),(9,96,19,50),(10,91,20,57),(11,86,21,52),(12,93,22,59),(25,74,67,41),(26,81,68,48),(27,76,69,43),(28,83,70,38),(29,78,71,45),(30,73,72,40),(31,80,61,47),(32,75,62,42),(33,82,63,37),(34,77,64,44),(35,84,65,39),(36,79,66,46)], [(1,62,7,68),(2,27,8,33),(3,64,9,70),(4,29,10,35),(5,66,11,72),(6,31,12,25),(13,34,19,28),(14,71,20,65),(15,36,21,30),(16,61,22,67),(17,26,23,32),(18,63,24,69),(37,49,43,55),(38,90,44,96),(39,51,45,57),(40,92,46,86),(41,53,47,59),(42,94,48,88),(50,83,56,77),(52,73,58,79),(54,75,60,81),(74,87,80,93),(76,89,82,95),(78,91,84,85)], [(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)], [(1,61,7,67),(2,66,8,72),(3,71,9,65),(4,64,10,70),(5,69,11,63),(6,62,12,68),(13,29,19,35),(14,34,20,28),(15,27,21,33),(16,32,22,26),(17,25,23,31),(18,30,24,36),(37,86,43,92),(38,91,44,85),(39,96,45,90),(40,89,46,95),(41,94,47,88),(42,87,48,93),(49,73,55,79),(50,78,56,84),(51,83,57,77),(52,76,58,82),(53,81,59,75),(54,74,60,80)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 8 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 8 | 0 |
G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,11,1,0],[1,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,5,8,0,0,0,0,0,5,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | 2- (1+4) | S3×C4○D4 | D4○D12 | Q8○D12 |
| kernel | C42.161D6 | C12.6Q8 | C4×D12 | Dic3.D4 | Dic3⋊4D4 | C23.9D6 | Dic3⋊D4 | C22.D12 | C12.3Q8 | S3×C4⋊C4 | D6.D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | D6 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{161}D_6 % in TeX
G:=Group("C4^2.161D6"); // GroupNames label
G:=SmallGroup(192,1266);
// by ID
G=gap.SmallGroup(192,1266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,570,80,6278]);
// 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*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations