metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.240D6, (C4×S3)⋊9D4, C4⋊Q8⋊19S3, D6.8(C2×D4), C4.38(S3×D4), C12⋊6(C4○D4), C4⋊C4.217D6, C12.70(C2×D4), C12⋊D4⋊39C2, C4⋊D12⋊17C2, (S3×C42)⋊14C2, C4⋊1(Q8⋊3S3), (C2×Q8).168D6, Dic3⋊5D4⋊43C2, C6.99(C22×D4), (C2×C6).269C24, D6⋊C4.50C22, Dic3.67(C2×D4), C12.23D4⋊26C2, (C2×C12).102C23, (C4×C12).210C22, (C6×Q8).136C22, (C2×D12).172C22, C22.290(S3×C23), C3⋊6(C22.26C24), (C22×S3).119C23, (C2×Dic3).272C23, (C4×Dic3).258C22, C2.72(C2×S3×D4), (C3×C4⋊Q8)⋊11C2, C6.120(C2×C4○D4), (C2×Q8⋊3S3)⋊12C2, (S3×C2×C4).143C22, C2.27(C2×Q8⋊3S3), (C3×C4⋊C4).212C22, (C2×C4).599(C22×S3), SmallGroup(192,1284)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 848 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×8], C22, C22 [×16], S3 [×6], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×19], D4 [×20], Q8 [×4], C23 [×5], Dic3 [×2], Dic3 [×2], C12 [×6], C12 [×4], D6 [×2], D6 [×14], C2×C6, C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C4×S3 [×4], C4×S3 [×12], D12 [×20], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], C4×Dic3, C4×Dic3 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×6], C2×D12 [×10], Q8⋊3S3 [×8], C6×Q8 [×2], C22.26C24, S3×C42, C4⋊D12, Dic3⋊5D4 [×4], C12⋊D4 [×4], C12.23D4 [×2], C3×C4⋊Q8, C2×Q8⋊3S3 [×2], C42.240D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], Q8⋊3S3 [×4], S3×C23, C22.26C24, C2×S3×D4, C2×Q8⋊3S3 [×2], C42.240D6
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c5 >
►On 96 points
(1 71 74 16)(2 17 75 72)(3 61 76 18)(4 19 77 62)(5 63 78 20)(6 21 79 64)(7 65 80 22)(8 23 81 66)(9 67 82 24)(10 13 83 68)(11 69 84 14)(12 15 73 70)(25 56 87 45)(26 46 88 57)(27 58 89 47)(28 48 90 59)(29 60 91 37)(30 38 92 49)(31 50 93 39)(32 40 94 51)(33 52 95 41)(34 42 96 53)(35 54 85 43)(36 44 86 55)
(1 32 7 26)(2 27 8 33)(3 34 9 28)(4 29 10 35)(5 36 11 30)(6 31 12 25)(13 54 19 60)(14 49 20 55)(15 56 21 50)(16 51 22 57)(17 58 23 52)(18 53 24 59)(37 68 43 62)(38 63 44 69)(39 70 45 64)(40 65 46 71)(41 72 47 66)(42 67 48 61)(73 87 79 93)(74 94 80 88)(75 89 81 95)(76 96 82 90)(77 91 83 85)(78 86 84 92)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 72)(24 71)(25 29)(26 28)(30 36)(31 35)(32 34)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)(73 83)(74 82)(75 81)(76 80)(77 79)(85 93)(86 92)(87 91)(88 90)(94 96)
G:=sub<Sym(96)| (1,71,74,16)(2,17,75,72)(3,61,76,18)(4,19,77,62)(5,63,78,20)(6,21,79,64)(7,65,80,22)(8,23,81,66)(9,67,82,24)(10,13,83,68)(11,69,84,14)(12,15,73,70)(25,56,87,45)(26,46,88,57)(27,58,89,47)(28,48,90,59)(29,60,91,37)(30,38,92,49)(31,50,93,39)(32,40,94,51)(33,52,95,41)(34,42,96,53)(35,54,85,43)(36,44,86,55), (1,32,7,26)(2,27,8,33)(3,34,9,28)(4,29,10,35)(5,36,11,30)(6,31,12,25)(13,54,19,60)(14,49,20,55)(15,56,21,50)(16,51,22,57)(17,58,23,52)(18,53,24,59)(37,68,43,62)(38,63,44,69)(39,70,45,64)(40,65,46,71)(41,72,47,66)(42,67,48,61)(73,87,79,93)(74,94,80,88)(75,89,81,95)(76,96,82,90)(77,91,83,85)(78,86,84,92), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,72)(24,71)(25,29)(26,28)(30,36)(31,35)(32,34)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(73,83)(74,82)(75,81)(76,80)(77,79)(85,93)(86,92)(87,91)(88,90)(94,96)>;
G:=Group( (1,71,74,16)(2,17,75,72)(3,61,76,18)(4,19,77,62)(5,63,78,20)(6,21,79,64)(7,65,80,22)(8,23,81,66)(9,67,82,24)(10,13,83,68)(11,69,84,14)(12,15,73,70)(25,56,87,45)(26,46,88,57)(27,58,89,47)(28,48,90,59)(29,60,91,37)(30,38,92,49)(31,50,93,39)(32,40,94,51)(33,52,95,41)(34,42,96,53)(35,54,85,43)(36,44,86,55), (1,32,7,26)(2,27,8,33)(3,34,9,28)(4,29,10,35)(5,36,11,30)(6,31,12,25)(13,54,19,60)(14,49,20,55)(15,56,21,50)(16,51,22,57)(17,58,23,52)(18,53,24,59)(37,68,43,62)(38,63,44,69)(39,70,45,64)(40,65,46,71)(41,72,47,66)(42,67,48,61)(73,87,79,93)(74,94,80,88)(75,89,81,95)(76,96,82,90)(77,91,83,85)(78,86,84,92), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,72)(24,71)(25,29)(26,28)(30,36)(31,35)(32,34)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(73,83)(74,82)(75,81)(76,80)(77,79)(85,93)(86,92)(87,91)(88,90)(94,96) );
G=PermutationGroup([(1,71,74,16),(2,17,75,72),(3,61,76,18),(4,19,77,62),(5,63,78,20),(6,21,79,64),(7,65,80,22),(8,23,81,66),(9,67,82,24),(10,13,83,68),(11,69,84,14),(12,15,73,70),(25,56,87,45),(26,46,88,57),(27,58,89,47),(28,48,90,59),(29,60,91,37),(30,38,92,49),(31,50,93,39),(32,40,94,51),(33,52,95,41),(34,42,96,53),(35,54,85,43),(36,44,86,55)], [(1,32,7,26),(2,27,8,33),(3,34,9,28),(4,29,10,35),(5,36,11,30),(6,31,12,25),(13,54,19,60),(14,49,20,55),(15,56,21,50),(16,51,22,57),(17,58,23,52),(18,53,24,59),(37,68,43,62),(38,63,44,69),(39,70,45,64),(40,65,46,71),(41,72,47,66),(42,67,48,61),(73,87,79,93),(74,94,80,88),(75,89,81,95),(76,96,82,90),(77,91,83,85),(78,86,84,92)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,72),(24,71),(25,29),(26,28),(30,36),(31,35),(32,34),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57),(73,83),(74,82),(75,81),(76,80),(77,79),(85,93),(86,92),(87,91),(88,90),(94,96)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [5,3,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,10,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | S3×D4 | Q8⋊3S3 |
| kernel | C42.240D6 | S3×C42 | C4⋊D12 | Dic3⋊5D4 | C12⋊D4 | C12.23D4 | C3×C4⋊Q8 | C2×Q8⋊3S3 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{240}D_6 % in TeX
G:=Group("C4^2.240D6"); // GroupNames label
G:=SmallGroup(192,1284);
// by ID
G=gap.SmallGroup(192,1284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^5>;
// generators/relations