metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.93D6, (C4×D12)⋊9C2, C4⋊C4.270D6, (C4×Dic6)⋊9C2, (S3×C42)⋊18C2, D6⋊Q8⋊50C2, D6.1(C4○D4), Dic3⋊D4.6C2, (C2×C6).72C24, C22⋊C4.96D6, C42⋊2S3⋊30C2, C42⋊C2⋊12S3, C23.9D6⋊52C2, D6.D4⋊48C2, Dic3.Q8⋊44C2, (C22×C4).209D6, C12.255(C4○D4), C4.139(C4○D12), (C4×C12).233C22, (C2×C12).147C23, D6⋊C4.143C22, Dic3.2(C4○D4), C23.8D6⋊48C2, C23.94(C22×S3), C23.11D6⋊48C2, (C2×D12).207C22, C4⋊Dic3.292C22, (C22×C6).142C23, C22.101(S3×C23), (C2×Dic3).25C23, Dic3⋊C4.152C22, (C22×S3).165C23, (C22×C12).377C22, C3⋊2(C23.36C23), (C2×Dic6).230C22, (C4×Dic3).196C22, C6.D4.95C22, (C4×C3⋊D4)⋊52C2, C4⋊C4⋊S3⋊49C2, C2.11(S3×C4○D4), C6.29(C2×C4○D4), C2.31(C2×C4○D12), (S3×C2×C4).290C22, (C3×C42⋊C2)⋊14C2, (C3×C4⋊C4).308C22, (C2×C4).150(C22×S3), (C2×C3⋊D4).102C22, (C3×C22⋊C4).112C22, SmallGroup(192,1087)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 552 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×5], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic6 [×2], C4×S3 [×8], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×2], C22×S3 [×2], C22×C6, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×4], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×4], C2×D12, C2×C3⋊D4 [×2], C22×C12, C23.36C23, C4×Dic6, S3×C42, C42⋊2S3, C4×D12, C23.8D6, C23.9D6, Dic3⋊D4, C23.11D6, Dic3.Q8, D6.D4, D6⋊Q8, C4⋊C4⋊S3, C4×C3⋊D4 [×2], C3×C42⋊C2, C42.93D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], C4○D12 [×2], S3×C23, C23.36C23, C2×C4○D12, S3×C4○D4 [×2], C42.93D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 79 40 85)(2 80 41 86)(3 81 42 87)(4 82 43 88)(5 83 44 89)(6 84 45 90)(7 73 46 91)(8 74 47 92)(9 75 48 93)(10 76 37 94)(11 77 38 95)(12 78 39 96)(13 50 34 64)(14 51 35 65)(15 52 36 66)(16 53 25 67)(17 54 26 68)(18 55 27 69)(19 56 28 70)(20 57 29 71)(21 58 30 72)(22 59 31 61)(23 60 32 62)(24 49 33 63)
(1 29 46 14)(2 21 47 36)(3 31 48 16)(4 23 37 26)(5 33 38 18)(6 13 39 28)(7 35 40 20)(8 15 41 30)(9 25 42 22)(10 17 43 32)(11 27 44 24)(12 19 45 34)(49 77 69 89)(50 96 70 84)(51 79 71 91)(52 86 72 74)(53 81 61 93)(54 88 62 76)(55 83 63 95)(56 90 64 78)(57 73 65 85)(58 92 66 80)(59 75 67 87)(60 94 68 82)
(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 70 7 64)(2 63 8 69)(3 68 9 62)(4 61 10 67)(5 66 11 72)(6 71 12 65)(13 79 19 73)(14 84 20 78)(15 77 21 83)(16 82 22 76)(17 75 23 81)(18 80 24 74)(25 88 31 94)(26 93 32 87)(27 86 33 92)(28 91 34 85)(29 96 35 90)(30 89 36 95)(37 53 43 59)(38 58 44 52)(39 51 45 57)(40 56 46 50)(41 49 47 55)(42 54 48 60)
G:=sub<Sym(96)| (1,79,40,85)(2,80,41,86)(3,81,42,87)(4,82,43,88)(5,83,44,89)(6,84,45,90)(7,73,46,91)(8,74,47,92)(9,75,48,93)(10,76,37,94)(11,77,38,95)(12,78,39,96)(13,50,34,64)(14,51,35,65)(15,52,36,66)(16,53,25,67)(17,54,26,68)(18,55,27,69)(19,56,28,70)(20,57,29,71)(21,58,30,72)(22,59,31,61)(23,60,32,62)(24,49,33,63), (1,29,46,14)(2,21,47,36)(3,31,48,16)(4,23,37,26)(5,33,38,18)(6,13,39,28)(7,35,40,20)(8,15,41,30)(9,25,42,22)(10,17,43,32)(11,27,44,24)(12,19,45,34)(49,77,69,89)(50,96,70,84)(51,79,71,91)(52,86,72,74)(53,81,61,93)(54,88,62,76)(55,83,63,95)(56,90,64,78)(57,73,65,85)(58,92,66,80)(59,75,67,87)(60,94,68,82), (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), (1,70,7,64)(2,63,8,69)(3,68,9,62)(4,61,10,67)(5,66,11,72)(6,71,12,65)(13,79,19,73)(14,84,20,78)(15,77,21,83)(16,82,22,76)(17,75,23,81)(18,80,24,74)(25,88,31,94)(26,93,32,87)(27,86,33,92)(28,91,34,85)(29,96,35,90)(30,89,36,95)(37,53,43,59)(38,58,44,52)(39,51,45,57)(40,56,46,50)(41,49,47,55)(42,54,48,60)>;
G:=Group( (1,79,40,85)(2,80,41,86)(3,81,42,87)(4,82,43,88)(5,83,44,89)(6,84,45,90)(7,73,46,91)(8,74,47,92)(9,75,48,93)(10,76,37,94)(11,77,38,95)(12,78,39,96)(13,50,34,64)(14,51,35,65)(15,52,36,66)(16,53,25,67)(17,54,26,68)(18,55,27,69)(19,56,28,70)(20,57,29,71)(21,58,30,72)(22,59,31,61)(23,60,32,62)(24,49,33,63), (1,29,46,14)(2,21,47,36)(3,31,48,16)(4,23,37,26)(5,33,38,18)(6,13,39,28)(7,35,40,20)(8,15,41,30)(9,25,42,22)(10,17,43,32)(11,27,44,24)(12,19,45,34)(49,77,69,89)(50,96,70,84)(51,79,71,91)(52,86,72,74)(53,81,61,93)(54,88,62,76)(55,83,63,95)(56,90,64,78)(57,73,65,85)(58,92,66,80)(59,75,67,87)(60,94,68,82), (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), (1,70,7,64)(2,63,8,69)(3,68,9,62)(4,61,10,67)(5,66,11,72)(6,71,12,65)(13,79,19,73)(14,84,20,78)(15,77,21,83)(16,82,22,76)(17,75,23,81)(18,80,24,74)(25,88,31,94)(26,93,32,87)(27,86,33,92)(28,91,34,85)(29,96,35,90)(30,89,36,95)(37,53,43,59)(38,58,44,52)(39,51,45,57)(40,56,46,50)(41,49,47,55)(42,54,48,60) );
G=PermutationGroup([(1,79,40,85),(2,80,41,86),(3,81,42,87),(4,82,43,88),(5,83,44,89),(6,84,45,90),(7,73,46,91),(8,74,47,92),(9,75,48,93),(10,76,37,94),(11,77,38,95),(12,78,39,96),(13,50,34,64),(14,51,35,65),(15,52,36,66),(16,53,25,67),(17,54,26,68),(18,55,27,69),(19,56,28,70),(20,57,29,71),(21,58,30,72),(22,59,31,61),(23,60,32,62),(24,49,33,63)], [(1,29,46,14),(2,21,47,36),(3,31,48,16),(4,23,37,26),(5,33,38,18),(6,13,39,28),(7,35,40,20),(8,15,41,30),(9,25,42,22),(10,17,43,32),(11,27,44,24),(12,19,45,34),(49,77,69,89),(50,96,70,84),(51,79,71,91),(52,86,72,74),(53,81,61,93),(54,88,62,76),(55,83,63,95),(56,90,64,78),(57,73,65,85),(58,92,66,80),(59,75,67,87),(60,94,68,82)], [(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)], [(1,70,7,64),(2,63,8,69),(3,68,9,62),(4,61,10,67),(5,66,11,72),(6,71,12,65),(13,79,19,73),(14,84,20,78),(15,77,21,83),(16,82,22,76),(17,75,23,81),(18,80,24,74),(25,88,31,94),(26,93,32,87),(27,86,33,92),(28,91,34,85),(29,96,35,90),(30,89,36,95),(37,53,43,59),(38,58,44,52),(39,51,45,57),(40,56,46,50),(41,49,47,55),(42,54,48,60)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 11 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,11,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | C4○D12 | S3×C4○D4 |
| kernel | C42.93D6 | C4×Dic6 | S3×C42 | C42⋊2S3 | C4×D12 | C23.8D6 | C23.9D6 | Dic3⋊D4 | C23.11D6 | Dic3.Q8 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{93}D_6 % in TeX
G:=Group("C4^2.93D6"); // GroupNames label
G:=SmallGroup(192,1087);
// by ID
G=gap.SmallGroup(192,1087);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations