direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C12×C4○D4, D4⋊6(C2×C12), (C4×D4)⋊23C6, (C4×Q8)⋊22C6, Q8⋊7(C2×C12), (D4×C12)⋊52C2, (C2×C42)⋊11C6, (Q8×C12)⋊38C2, C42.87(C2×C6), C6.58(C23×C4), C2.6(C23×C12), C42⋊C2⋊19C6, (C2×C6).337C24, C4.18(C22×C12), C12.163(C22×C4), (C2×C12).959C23, (C4×C12).371C22, (C6×D4).331C22, C22.1(C22×C12), C23.33(C22×C6), C22.10(C23×C6), (C6×Q8).283C22, (C22×C6).253C23, (C22×C12).595C22, (C2×C4×C12)⋊21C2, (C2×C4)⋊8(C2×C12), C2.4(C6×C4○D4), (C2×C12)⋊33(C2×C4), (C3×D4)⋊26(C2×C4), C4⋊C4.81(C2×C6), (C3×Q8)⋊24(C2×C4), (C6×C4○D4).27C2, (C2×C4○D4).19C6, (C2×D4).77(C2×C6), C6.223(C2×C4○D4), (C2×Q8).83(C2×C6), C22⋊C4.28(C2×C6), (C2×C6).32(C22×C4), (C3×C42⋊C2)⋊40C2, (C3×C4⋊C4).406C22, (C2×C4).134(C22×C6), (C22×C4).106(C2×C6), (C3×C22⋊C4).159C22, SmallGroup(192,1406)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 370 in 310 conjugacy classes, 250 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C12 [×12], C12 [×6], C2×C6, C2×C6 [×6], C2×C6 [×6], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C12, C2×C12 [×23], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C4×C12, C4×C12 [×9], C3×C22⋊C4 [×6], C3×C4⋊C4 [×6], C22×C12 [×9], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C4×C4○D4, C2×C4×C12 [×3], C3×C42⋊C2 [×3], D4×C12 [×6], Q8×C12 [×2], C6×C4○D4, C12×C4○D4
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C4○D4 [×4], C24, C2×C12 [×28], C22×C6 [×15], C23×C4, C2×C4○D4 [×2], C22×C12 [×14], C3×C4○D4 [×4], C23×C6, C4×C4○D4, C23×C12, C6×C4○D4 [×2], C12×C4○D4
Generators and relations
G = < a,b,c,d | a12=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
►On 96 points
(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 15 39 66)(2 16 40 67)(3 17 41 68)(4 18 42 69)(5 19 43 70)(6 20 44 71)(7 21 45 72)(8 22 46 61)(9 23 47 62)(10 24 48 63)(11 13 37 64)(12 14 38 65)(25 74 90 51)(26 75 91 52)(27 76 92 53)(28 77 93 54)(29 78 94 55)(30 79 95 56)(31 80 96 57)(32 81 85 58)(33 82 86 59)(34 83 87 60)(35 84 88 49)(36 73 89 50)
(1 74 39 51)(2 75 40 52)(3 76 41 53)(4 77 42 54)(5 78 43 55)(6 79 44 56)(7 80 45 57)(8 81 46 58)(9 82 47 59)(10 83 48 60)(11 84 37 49)(12 73 38 50)(13 88 64 35)(14 89 65 36)(15 90 66 25)(16 91 67 26)(17 92 68 27)(18 93 69 28)(19 94 70 29)(20 95 71 30)(21 96 72 31)(22 85 61 32)(23 86 62 33)(24 87 63 34)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 96)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 73)(57 74)(58 75)(59 76)(60 77)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
G:=sub<Sym(96)| (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), (1,15,39,66)(2,16,40,67)(3,17,41,68)(4,18,42,69)(5,19,43,70)(6,20,44,71)(7,21,45,72)(8,22,46,61)(9,23,47,62)(10,24,48,63)(11,13,37,64)(12,14,38,65)(25,74,90,51)(26,75,91,52)(27,76,92,53)(28,77,93,54)(29,78,94,55)(30,79,95,56)(31,80,96,57)(32,81,85,58)(33,82,86,59)(34,83,87,60)(35,84,88,49)(36,73,89,50), (1,74,39,51)(2,75,40,52)(3,76,41,53)(4,77,42,54)(5,78,43,55)(6,79,44,56)(7,80,45,57)(8,81,46,58)(9,82,47,59)(10,83,48,60)(11,84,37,49)(12,73,38,50)(13,88,64,35)(14,89,65,36)(15,90,66,25)(16,91,67,26)(17,92,68,27)(18,93,69,28)(19,94,70,29)(20,95,71,30)(21,96,72,31)(22,85,61,32)(23,86,62,33)(24,87,63,34), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)>;
G:=Group( (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), (1,15,39,66)(2,16,40,67)(3,17,41,68)(4,18,42,69)(5,19,43,70)(6,20,44,71)(7,21,45,72)(8,22,46,61)(9,23,47,62)(10,24,48,63)(11,13,37,64)(12,14,38,65)(25,74,90,51)(26,75,91,52)(27,76,92,53)(28,77,93,54)(29,78,94,55)(30,79,95,56)(31,80,96,57)(32,81,85,58)(33,82,86,59)(34,83,87,60)(35,84,88,49)(36,73,89,50), (1,74,39,51)(2,75,40,52)(3,76,41,53)(4,77,42,54)(5,78,43,55)(6,79,44,56)(7,80,45,57)(8,81,46,58)(9,82,47,59)(10,83,48,60)(11,84,37,49)(12,73,38,50)(13,88,64,35)(14,89,65,36)(15,90,66,25)(16,91,67,26)(17,92,68,27)(18,93,69,28)(19,94,70,29)(20,95,71,30)(21,96,72,31)(22,85,61,32)(23,86,62,33)(24,87,63,34), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72) );
G=PermutationGroup([(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)], [(1,15,39,66),(2,16,40,67),(3,17,41,68),(4,18,42,69),(5,19,43,70),(6,20,44,71),(7,21,45,72),(8,22,46,61),(9,23,47,62),(10,24,48,63),(11,13,37,64),(12,14,38,65),(25,74,90,51),(26,75,91,52),(27,76,92,53),(28,77,93,54),(29,78,94,55),(30,79,95,56),(31,80,96,57),(32,81,85,58),(33,82,86,59),(34,83,87,60),(35,84,88,49),(36,73,89,50)], [(1,74,39,51),(2,75,40,52),(3,76,41,53),(4,77,42,54),(5,78,43,55),(6,79,44,56),(7,80,45,57),(8,81,46,58),(9,82,47,59),(10,83,48,60),(11,84,37,49),(12,73,38,50),(13,88,64,35),(14,89,65,36),(15,90,66,25),(16,91,67,26),(17,92,68,27),(18,93,69,28),(19,94,70,29),(20,95,71,30),(21,96,72,31),(22,85,61,32),(23,86,62,33),(24,87,63,34)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,96),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,73),(57,74),(58,75),(59,76),(60,77),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)])
Matrix representation ►G ⊆ GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 12 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 10 | 5 |
| 0 | 11 | 3 |
| 1 | 0 | 0 |
| 0 | 12 | 3 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [8,0,0,0,11,0,0,0,11],[12,0,0,0,5,0,0,0,5],[1,0,0,0,10,11,0,5,3],[1,0,0,0,12,0,0,3,1] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4AD | 6A | ··· | 6F | 6G | ··· | 6R | 12A | ··· | 12X | 12Y | ··· | 12BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C4○D4 | C3×C4○D4 |
| kernel | C12×C4○D4 | C2×C4×C12 | C3×C42⋊C2 | D4×C12 | Q8×C12 | C6×C4○D4 | C4×C4○D4 | C3×C4○D4 | C2×C42 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C12 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 2 | 16 | 6 | 6 | 12 | 4 | 2 | 32 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_{12}\times C_4\circ D_4 % in TeX
G:=Group("C12xC4oD4"); // GroupNames label
G:=SmallGroup(192,1406);
// by ID
G=gap.SmallGroup(192,1406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,520,192]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations