metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.97D6, C6.982+ (1+4), C12⋊Q8⋊12C2, Dic3⋊D4⋊4C2, C4⋊C4.311D6, C12⋊D4⋊13C2, C12⋊7D4⋊30C2, C42⋊2S3⋊4C2, C42⋊7S3⋊7C2, (C2×C6).76C24, Dic3⋊5D4⋊13C2, C42⋊C2⋊16S3, C2.10(D4○D12), C4.97(C4○D12), (C4×C12).27C22, D6⋊C4.83C22, C22⋊C4.100D6, (C22×C4).213D6, C12.199(C4○D4), (C2×C12).697C23, C23.11D6⋊4C2, (C2×D12).24C22, C23.97(C22×S3), Dic3.18(C4○D4), (C22×S3).24C23, C4⋊Dic3.196C22, (C22×C6).146C23, C22.105(S3×C23), (C2×Dic3).29C23, (C2×Dic6).24C22, Dic3⋊C4.106C22, (C22×C12).233C22, C3⋊1(C22.49C24), (C4×Dic3).198C22, C6.D4.98C22, (C4×C3⋊D4)⋊13C2, C4⋊C4⋊7S3⋊12C2, C2.15(S3×C4○D4), C6.32(C2×C4○D4), C2.35(C2×C4○D12), (S3×C2×C4).194C22, (C3×C42⋊C2)⋊18C2, (C3×C4⋊C4).312C22, (C2×C4).278(C22×S3), (C2×C3⋊D4).105C22, (C3×C22⋊C4).115C22, SmallGroup(192,1091)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×11], C22, C22 [×12], S3 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×8], Q8 [×2], C23, C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×5], D6 [×9], C2×C6, C2×C6 [×3], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×6], C2×Q8 [×2], Dic6 [×2], C4×S3 [×6], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3, C22×S3 [×2], C22×C6, C42⋊C2, C42⋊C2 [×3], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4, D6⋊C4 [×8], C6.D4, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C2×C3⋊D4, C2×C3⋊D4 [×2], C22×C12, C22.49C24, C42⋊2S3 [×2], C42⋊7S3 [×2], Dic3⋊D4 [×2], C23.11D6 [×2], C12⋊Q8, C4⋊C4⋊7S3, Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C12⋊7D4, C3×C42⋊C2, C42.97D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), C4○D12 [×2], S3×C23, C22.49C24, C2×C4○D12, S3×C4○D4, D4○D12, C42.97D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 65 83 60)(2 72 84 55)(3 67 73 50)(4 62 74 57)(5 69 75 52)(6 64 76 59)(7 71 77 54)(8 66 78 49)(9 61 79 56)(10 68 80 51)(11 63 81 58)(12 70 82 53)(13 32 92 37)(14 27 93 44)(15 34 94 39)(16 29 95 46)(17 36 96 41)(18 31 85 48)(19 26 86 43)(20 33 87 38)(21 28 88 45)(22 35 89 40)(23 30 90 47)(24 25 91 42)
(1 26 7 32)(2 27 8 33)(3 28 9 34)(4 29 10 35)(5 30 11 36)(6 31 12 25)(13 60 19 54)(14 49 20 55)(15 50 21 56)(16 51 22 57)(17 52 23 58)(18 53 24 59)(37 83 43 77)(38 84 44 78)(39 73 45 79)(40 74 46 80)(41 75 47 81)(42 76 48 82)(61 94 67 88)(62 95 68 89)(63 96 69 90)(64 85 70 91)(65 86 71 92)(66 87 72 93)
(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 21 7 15)(2 14 8 20)(3 19 9 13)(4 24 10 18)(5 17 11 23)(6 22 12 16)(25 68 31 62)(26 61 32 67)(27 66 33 72)(28 71 34 65)(29 64 35 70)(30 69 36 63)(37 50 43 56)(38 55 44 49)(39 60 45 54)(40 53 46 59)(41 58 47 52)(42 51 48 57)(73 86 79 92)(74 91 80 85)(75 96 81 90)(76 89 82 95)(77 94 83 88)(78 87 84 93)
G:=sub<Sym(96)| (1,65,83,60)(2,72,84,55)(3,67,73,50)(4,62,74,57)(5,69,75,52)(6,64,76,59)(7,71,77,54)(8,66,78,49)(9,61,79,56)(10,68,80,51)(11,63,81,58)(12,70,82,53)(13,32,92,37)(14,27,93,44)(15,34,94,39)(16,29,95,46)(17,36,96,41)(18,31,85,48)(19,26,86,43)(20,33,87,38)(21,28,88,45)(22,35,89,40)(23,30,90,47)(24,25,91,42), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,60,19,54)(14,49,20,55)(15,50,21,56)(16,51,22,57)(17,52,23,58)(18,53,24,59)(37,83,43,77)(38,84,44,78)(39,73,45,79)(40,74,46,80)(41,75,47,81)(42,76,48,82)(61,94,67,88)(62,95,68,89)(63,96,69,90)(64,85,70,91)(65,86,71,92)(66,87,72,93), (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), (1,21,7,15)(2,14,8,20)(3,19,9,13)(4,24,10,18)(5,17,11,23)(6,22,12,16)(25,68,31,62)(26,61,32,67)(27,66,33,72)(28,71,34,65)(29,64,35,70)(30,69,36,63)(37,50,43,56)(38,55,44,49)(39,60,45,54)(40,53,46,59)(41,58,47,52)(42,51,48,57)(73,86,79,92)(74,91,80,85)(75,96,81,90)(76,89,82,95)(77,94,83,88)(78,87,84,93)>;
G:=Group( (1,65,83,60)(2,72,84,55)(3,67,73,50)(4,62,74,57)(5,69,75,52)(6,64,76,59)(7,71,77,54)(8,66,78,49)(9,61,79,56)(10,68,80,51)(11,63,81,58)(12,70,82,53)(13,32,92,37)(14,27,93,44)(15,34,94,39)(16,29,95,46)(17,36,96,41)(18,31,85,48)(19,26,86,43)(20,33,87,38)(21,28,88,45)(22,35,89,40)(23,30,90,47)(24,25,91,42), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,60,19,54)(14,49,20,55)(15,50,21,56)(16,51,22,57)(17,52,23,58)(18,53,24,59)(37,83,43,77)(38,84,44,78)(39,73,45,79)(40,74,46,80)(41,75,47,81)(42,76,48,82)(61,94,67,88)(62,95,68,89)(63,96,69,90)(64,85,70,91)(65,86,71,92)(66,87,72,93), (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), (1,21,7,15)(2,14,8,20)(3,19,9,13)(4,24,10,18)(5,17,11,23)(6,22,12,16)(25,68,31,62)(26,61,32,67)(27,66,33,72)(28,71,34,65)(29,64,35,70)(30,69,36,63)(37,50,43,56)(38,55,44,49)(39,60,45,54)(40,53,46,59)(41,58,47,52)(42,51,48,57)(73,86,79,92)(74,91,80,85)(75,96,81,90)(76,89,82,95)(77,94,83,88)(78,87,84,93) );
G=PermutationGroup([(1,65,83,60),(2,72,84,55),(3,67,73,50),(4,62,74,57),(5,69,75,52),(6,64,76,59),(7,71,77,54),(8,66,78,49),(9,61,79,56),(10,68,80,51),(11,63,81,58),(12,70,82,53),(13,32,92,37),(14,27,93,44),(15,34,94,39),(16,29,95,46),(17,36,96,41),(18,31,85,48),(19,26,86,43),(20,33,87,38),(21,28,88,45),(22,35,89,40),(23,30,90,47),(24,25,91,42)], [(1,26,7,32),(2,27,8,33),(3,28,9,34),(4,29,10,35),(5,30,11,36),(6,31,12,25),(13,60,19,54),(14,49,20,55),(15,50,21,56),(16,51,22,57),(17,52,23,58),(18,53,24,59),(37,83,43,77),(38,84,44,78),(39,73,45,79),(40,74,46,80),(41,75,47,81),(42,76,48,82),(61,94,67,88),(62,95,68,89),(63,96,69,90),(64,85,70,91),(65,86,71,92),(66,87,72,93)], [(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)], [(1,21,7,15),(2,14,8,20),(3,19,9,13),(4,24,10,18),(5,17,11,23),(6,22,12,16),(25,68,31,62),(26,61,32,67),(27,66,33,72),(28,71,34,65),(29,64,35,70),(30,69,36,63),(37,50,43,56),(38,55,44,49),(39,60,45,54),(40,53,46,59),(41,58,47,52),(42,51,48,57),(73,86,79,92),(74,91,80,85),(75,96,81,90),(76,89,82,95),(77,94,83,88),(78,87,84,93)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 12 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 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 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ (1+4) | S3×C4○D4 | D4○D12 |
| kernel | C42.97D6 | C42⋊2S3 | C42⋊7S3 | Dic3⋊D4 | C23.11D6 | C12⋊Q8 | C4⋊C4⋊7S3 | Dic3⋊5D4 | C12⋊D4 | C4×C3⋊D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{97}D_6 % in TeX
G:=Group("C4^2.97D6"); // GroupNames label
G:=SmallGroup(192,1091);
// by ID
G=gap.SmallGroup(192,1091);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,675,136,6278]);
// 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*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations