direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.15D6, C6.10C25, D6.5C24, C12.45C24, C6⋊12- (1+4), D12.36C23, Dic3.5C24, Dic6.36C23, (C2×Q8)⋊38D6, C4.45(S3×C23), C2.11(S3×C24), (S3×Q8)⋊13C22, (C22×Q8)⋊17S3, (C6×Q8)⋊44C22, C3⋊D4.6C23, C3⋊1(C2×2- (1+4)), C4○D12⋊23C22, (C4×S3).18C23, (C2×C6).330C24, (C22×C4).307D6, C22.9(S3×C23), (C3×Q8).29C23, Q8.39(C22×S3), (C2×C12).566C23, Q8⋊3S3⋊12C22, (C2×D12).283C22, C23.252(C22×S3), (C22×C6).437C23, (C22×S3).248C23, (C22×C12).302C22, (C2×Dic6).312C22, (C2×Dic3).298C23, (Q8×C2×C6)⋊11C2, (C2×S3×Q8)⋊20C2, (C2×C4○D12)⋊35C2, (C2×Q8⋊3S3)⋊20C2, (S3×C2×C4).172C22, (C2×C4).252(C22×S3), (C2×C3⋊D4).150C22, SmallGroup(192,1519)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1512 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×12], C4 [×8], C22, C22 [×2], C22 [×18], S3 [×8], C6, C6 [×2], C6 [×2], C2×C4 [×18], C2×C4 [×52], D4 [×40], Q8 [×16], Q8 [×24], C23, C23 [×4], Dic3 [×8], C12 [×12], D6 [×8], D6 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C22×C4 [×3], C22×C4 [×12], C2×D4 [×10], C2×Q8 [×12], C2×Q8 [×38], C4○D4 [×80], Dic6 [×24], C4×S3 [×48], D12 [×24], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×18], C3×Q8 [×16], C22×S3 [×4], C22×C6, C22×Q8, C22×Q8 [×4], C2×C4○D4 [×10], 2- (1+4) [×16], C2×Dic6 [×6], S3×C2×C4 [×12], C2×D12 [×6], C4○D12 [×48], S3×Q8 [×32], Q8⋊3S3 [×32], C2×C3⋊D4 [×4], C22×C12 [×3], C6×Q8 [×12], C2×2- (1+4), C2×C4○D12 [×6], C2×S3×Q8 [×4], C2×Q8⋊3S3 [×4], Q8.15D6 [×16], Q8×C2×C6, C2×Q8.15D6
Quotients:
C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], 2- (1+4) [×2], C25, S3×C23 [×15], C2×2- (1+4), Q8.15D6 [×2], S3×C24, C2×Q8.15D6
Generators and relations
G = < a,b,c,d,e | a2=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d5 >
►On 96 points
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 61)(45 62)(46 63)(47 64)(48 65)(49 80)(50 81)(51 82)(52 83)(53 84)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)
(1 59 7 53)(2 60 8 54)(3 49 9 55)(4 50 10 56)(5 51 11 57)(6 52 12 58)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)(25 76 31 82)(26 77 32 83)(27 78 33 84)(28 79 34 73)(29 80 35 74)(30 81 36 75)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 90)
(1 69 7 63)(2 64 8 70)(3 71 9 65)(4 66 10 72)(5 61 11 67)(6 68 12 62)(13 75 19 81)(14 82 20 76)(15 77 21 83)(16 84 22 78)(17 79 23 73)(18 74 24 80)(25 38 31 44)(26 45 32 39)(27 40 33 46)(28 47 34 41)(29 42 35 48)(30 37 36 43)(49 95 55 89)(50 90 56 96)(51 85 57 91)(52 92 58 86)(53 87 59 93)(54 94 60 88)
(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 58 7 52)(2 51 8 57)(3 56 9 50)(4 49 10 55)(5 54 11 60)(6 59 12 53)(13 48 19 42)(14 41 20 47)(15 46 21 40)(16 39 22 45)(17 44 23 38)(18 37 24 43)(25 79 31 73)(26 84 32 78)(27 77 33 83)(28 82 34 76)(29 75 35 81)(30 80 36 74)(61 88 67 94)(62 93 68 87)(63 86 69 92)(64 91 70 85)(65 96 71 90)(66 89 72 95)
G:=sub<Sym(96)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,61)(45,62)(46,63)(47,64)(48,65)(49,80)(50,81)(51,82)(52,83)(53,84)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (1,59,7,53)(2,60,8,54)(3,49,9,55)(4,50,10,56)(5,51,11,57)(6,52,12,58)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(61,91,67,85)(62,92,68,86)(63,93,69,87)(64,94,70,88)(65,95,71,89)(66,96,72,90), (1,69,7,63)(2,64,8,70)(3,71,9,65)(4,66,10,72)(5,61,11,67)(6,68,12,62)(13,75,19,81)(14,82,20,76)(15,77,21,83)(16,84,22,78)(17,79,23,73)(18,74,24,80)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(49,95,55,89)(50,90,56,96)(51,85,57,91)(52,92,58,86)(53,87,59,93)(54,94,60,88), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,48,19,42)(14,41,20,47)(15,46,21,40)(16,39,22,45)(17,44,23,38)(18,37,24,43)(25,79,31,73)(26,84,32,78)(27,77,33,83)(28,82,34,76)(29,75,35,81)(30,80,36,74)(61,88,67,94)(62,93,68,87)(63,86,69,92)(64,91,70,85)(65,96,71,90)(66,89,72,95)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,61)(45,62)(46,63)(47,64)(48,65)(49,80)(50,81)(51,82)(52,83)(53,84)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (1,59,7,53)(2,60,8,54)(3,49,9,55)(4,50,10,56)(5,51,11,57)(6,52,12,58)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(61,91,67,85)(62,92,68,86)(63,93,69,87)(64,94,70,88)(65,95,71,89)(66,96,72,90), (1,69,7,63)(2,64,8,70)(3,71,9,65)(4,66,10,72)(5,61,11,67)(6,68,12,62)(13,75,19,81)(14,82,20,76)(15,77,21,83)(16,84,22,78)(17,79,23,73)(18,74,24,80)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(49,95,55,89)(50,90,56,96)(51,85,57,91)(52,92,58,86)(53,87,59,93)(54,94,60,88), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,48,19,42)(14,41,20,47)(15,46,21,40)(16,39,22,45)(17,44,23,38)(18,37,24,43)(25,79,31,73)(26,84,32,78)(27,77,33,83)(28,82,34,76)(29,75,35,81)(30,80,36,74)(61,88,67,94)(62,93,68,87)(63,86,69,92)(64,91,70,85)(65,96,71,90)(66,89,72,95) );
G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,61),(45,62),(46,63),(47,64),(48,65),(49,80),(50,81),(51,82),(52,83),(53,84),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79)], [(1,59,7,53),(2,60,8,54),(3,49,9,55),(4,50,10,56),(5,51,11,57),(6,52,12,58),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48),(25,76,31,82),(26,77,32,83),(27,78,33,84),(28,79,34,73),(29,80,35,74),(30,81,36,75),(61,91,67,85),(62,92,68,86),(63,93,69,87),(64,94,70,88),(65,95,71,89),(66,96,72,90)], [(1,69,7,63),(2,64,8,70),(3,71,9,65),(4,66,10,72),(5,61,11,67),(6,68,12,62),(13,75,19,81),(14,82,20,76),(15,77,21,83),(16,84,22,78),(17,79,23,73),(18,74,24,80),(25,38,31,44),(26,45,32,39),(27,40,33,46),(28,47,34,41),(29,42,35,48),(30,37,36,43),(49,95,55,89),(50,90,56,96),(51,85,57,91),(52,92,58,86),(53,87,59,93),(54,94,60,88)], [(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)], [(1,58,7,52),(2,51,8,57),(3,56,9,50),(4,49,10,55),(5,54,11,60),(6,59,12,53),(13,48,19,42),(14,41,20,47),(15,46,21,40),(16,39,22,45),(17,44,23,38),(18,37,24,43),(25,79,31,73),(26,84,32,78),(27,77,33,83),(28,82,34,76),(29,75,35,81),(30,80,36,74),(61,88,67,94),(62,93,68,87),(63,86,69,92),(64,91,70,85),(65,96,71,90),(66,89,72,95)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 12 | 4 | 0 |
| 0 | 0 | 1 | 6 | 0 | 4 |
| 0 | 0 | 4 | 0 | 6 | 1 |
| 0 | 0 | 0 | 4 | 12 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 8 | 8 |
| 0 | 0 | 3 | 0 | 5 | 3 |
| 0 | 0 | 8 | 8 | 10 | 3 |
| 0 | 0 | 5 | 3 | 10 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 2 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,4,0,0,0,12,6,0,4,0,0,4,0,6,12,0,0,0,4,1,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,3,3,8,5,0,0,10,0,8,3,0,0,8,5,10,10,0,0,8,3,3,0],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2M | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2- (1+4) | Q8.15D6 |
| kernel | C2×Q8.15D6 | C2×C4○D12 | C2×S3×Q8 | C2×Q8⋊3S3 | Q8.15D6 | Q8×C2×C6 | C22×Q8 | C22×C4 | C2×Q8 | C6 | C2 |
| # reps | 1 | 6 | 4 | 4 | 16 | 1 | 1 | 3 | 12 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8._{15}D_6 % in TeX
G:=Group("C2xQ8.15D6"); // GroupNames label
G:=SmallGroup(192,1519);
// by ID
G=gap.SmallGroup(192,1519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations