direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8○D12, C6.13C25, D6.7C24, C12.48C24, C6⋊22- (1+4), D12.40C23, Dic3.8C24, Dic6.37C23, C4○D4⋊21D6, (C2×C6).4C24, (C2×D4).254D6, C4.63(S3×C23), C2.14(S3×C24), (C2×Q8).236D6, (S3×Q8)⋊14C22, C3⋊D4.1C23, C3⋊2(C2×2- (1+4)), C4○D12⋊26C22, (C4×S3).19C23, (C3×D4).29C23, D4.29(C22×S3), (C22×C4).308D6, (C3×Q8).30C23, Q8.40(C22×S3), D4⋊2S3⋊13C22, (C2×C12).567C23, (C22×Dic6)⋊25C2, (C2×Dic6)⋊75C22, (C6×D4).279C22, C22.57(S3×C23), (C6×Q8).247C22, (C2×D12).290C22, (C22×C6).249C23, C23.225(C22×S3), (C22×S3).250C23, (C22×C12).303C22, (C2×Dic3).167C23, (C22×Dic3).169C22, (C2×S3×Q8)⋊21C2, (C6×C4○D4)⋊15C2, (C2×C4○D4)⋊18S3, (C2×C4○D12)⋊38C2, (C2×D4⋊2S3)⋊30C2, (C3×C4○D4)⋊21C22, (S3×C2×C4).173C22, (C2×C4).253(C22×S3), (C2×C3⋊D4).144C22, SmallGroup(192,1522)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1480 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C4 [×12], C22, C22 [×6], C22 [×14], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], C2×C4 [×54], D4 [×12], D4 [×28], Q8 [×4], Q8 [×36], C23 [×3], C23 [×2], Dic3 [×12], C12 [×8], D6 [×4], D6 [×4], C2×C6, C2×C6 [×6], C2×C6 [×6], C22×C4 [×3], C22×C4 [×12], C2×D4 [×3], C2×D4 [×7], C2×Q8, C2×Q8 [×49], C4○D4 [×8], C4○D4 [×72], Dic6 [×36], C4×S3 [×24], D12 [×4], C2×Dic3 [×30], C3⋊D4 [×24], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×2], C22×C6 [×3], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×9], 2- (1+4) [×16], C2×Dic6 [×33], S3×C2×C4 [×6], C2×D12, C4○D12 [×24], D4⋊2S3 [×48], S3×Q8 [×16], C22×Dic3 [×6], C2×C3⋊D4 [×6], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C2×2- (1+4), C22×Dic6 [×3], C2×C4○D12 [×3], C2×D4⋊2S3 [×6], C2×S3×Q8 [×2], Q8○D12 [×16], C6×C4○D4, C2×Q8○D12
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○D12 [×2], S3×C24, C2×Q8○D12
Generators and relations
G = < a,b,c,d,e | a2=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
►On 96 points
(1 58)(2 59)(3 60)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)
(1 62 7 68)(2 63 8 69)(3 64 9 70)(4 65 10 71)(5 66 11 72)(6 67 12 61)(13 51 19 57)(14 52 20 58)(15 53 21 59)(16 54 22 60)(17 55 23 49)(18 56 24 50)(25 88 31 94)(26 89 32 95)(27 90 33 96)(28 91 34 85)(29 92 35 86)(30 93 36 87)(37 77 43 83)(38 78 44 84)(39 79 45 73)(40 80 46 74)(41 81 47 75)(42 82 48 76)
(1 86 7 92)(2 87 8 93)(3 88 9 94)(4 89 10 95)(5 90 11 96)(6 91 12 85)(13 76 19 82)(14 77 20 83)(15 78 21 84)(16 79 22 73)(17 80 23 74)(18 81 24 75)(25 70 31 64)(26 71 32 65)(27 72 33 66)(28 61 34 67)(29 62 35 68)(30 63 36 69)(37 58 43 52)(38 59 44 53)(39 60 45 54)(40 49 46 55)(41 50 47 56)(42 51 48 57)
(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 51)(2 50)(3 49)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 72)(22 71)(23 70)(24 69)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 84)(34 83)(35 82)(36 81)(37 85)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)
G:=sub<Sym(96)| (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91), (1,62,7,68)(2,63,8,69)(3,64,9,70)(4,65,10,71)(5,66,11,72)(6,67,12,61)(13,51,19,57)(14,52,20,58)(15,53,21,59)(16,54,22,60)(17,55,23,49)(18,56,24,50)(25,88,31,94)(26,89,32,95)(27,90,33,96)(28,91,34,85)(29,92,35,86)(30,93,36,87)(37,77,43,83)(38,78,44,84)(39,79,45,73)(40,80,46,74)(41,81,47,75)(42,82,48,76), (1,86,7,92)(2,87,8,93)(3,88,9,94)(4,89,10,95)(5,90,11,96)(6,91,12,85)(13,76,19,82)(14,77,20,83)(15,78,21,84)(16,79,22,73)(17,80,23,74)(18,81,24,75)(25,70,31,64)(26,71,32,65)(27,72,33,66)(28,61,34,67)(29,62,35,68)(30,63,36,69)(37,58,43,52)(38,59,44,53)(39,60,45,54)(40,49,46,55)(41,50,47,56)(42,51,48,57), (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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(37,85)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)>;
G:=Group( (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91), (1,62,7,68)(2,63,8,69)(3,64,9,70)(4,65,10,71)(5,66,11,72)(6,67,12,61)(13,51,19,57)(14,52,20,58)(15,53,21,59)(16,54,22,60)(17,55,23,49)(18,56,24,50)(25,88,31,94)(26,89,32,95)(27,90,33,96)(28,91,34,85)(29,92,35,86)(30,93,36,87)(37,77,43,83)(38,78,44,84)(39,79,45,73)(40,80,46,74)(41,81,47,75)(42,82,48,76), (1,86,7,92)(2,87,8,93)(3,88,9,94)(4,89,10,95)(5,90,11,96)(6,91,12,85)(13,76,19,82)(14,77,20,83)(15,78,21,84)(16,79,22,73)(17,80,23,74)(18,81,24,75)(25,70,31,64)(26,71,32,65)(27,72,33,66)(28,61,34,67)(29,62,35,68)(30,63,36,69)(37,58,43,52)(38,59,44,53)(39,60,45,54)(40,49,46,55)(41,50,47,56)(42,51,48,57), (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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(37,85)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,92),(38,93),(39,94),(40,95),(41,96),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91)], [(1,62,7,68),(2,63,8,69),(3,64,9,70),(4,65,10,71),(5,66,11,72),(6,67,12,61),(13,51,19,57),(14,52,20,58),(15,53,21,59),(16,54,22,60),(17,55,23,49),(18,56,24,50),(25,88,31,94),(26,89,32,95),(27,90,33,96),(28,91,34,85),(29,92,35,86),(30,93,36,87),(37,77,43,83),(38,78,44,84),(39,79,45,73),(40,80,46,74),(41,81,47,75),(42,82,48,76)], [(1,86,7,92),(2,87,8,93),(3,88,9,94),(4,89,10,95),(5,90,11,96),(6,91,12,85),(13,76,19,82),(14,77,20,83),(15,78,21,84),(16,79,22,73),(17,80,23,74),(18,81,24,75),(25,70,31,64),(26,71,32,65),(27,72,33,66),(28,61,34,67),(29,62,35,68),(30,63,36,69),(37,58,43,52),(38,59,44,53),(39,60,45,54),(40,49,46,55),(41,50,47,56),(42,51,48,57)], [(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)], [(1,51),(2,50),(3,49),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,72),(22,71),(23,70),(24,69),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,84),(34,83),(35,82),(36,81),(37,85),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86)])
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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 2 | 0 |
| 0 | 0 | 6 | 10 | 0 | 2 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 10 | 0 |
| 0 | 0 | 4 | 11 | 0 | 10 |
| 0 | 0 | 5 | 0 | 11 | 4 |
| 0 | 0 | 0 | 5 | 9 | 2 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 10 | 6 |
| 0 | 0 | 1 | 12 | 3 | 3 |
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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,2,0,10,7,0,0,0,2,6,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,4,5,0,0,0,9,11,0,5,0,0,10,0,11,9,0,0,0,10,4,2],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,3,1,1,0,0,6,3,0,12,0,0,0,0,10,3,0,0,0,0,6,3] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 3 | 4A | ··· | 4H | 4I | ··· | 4T | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2- (1+4) | Q8○D12 |
| kernel | C2×Q8○D12 | C22×Dic6 | C2×C4○D12 | C2×D4⋊2S3 | C2×S3×Q8 | Q8○D12 | C6×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 1 | 3 | 3 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\circ D_{12} % in TeX
G:=Group("C2xQ8oD12"); // GroupNames label
G:=SmallGroup(192,1522);
// by ID
G=gap.SmallGroup(192,1522);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations