direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C12×SD16, (C4×C8)⋊11C6, C8⋊5(C2×C12), (C4×Q8)⋊5C6, C24⋊27(C2×C4), (C4×C24)⋊27C2, Q8⋊2(C2×C12), C4.Q8⋊14C6, (C4×D4).4C6, (Q8×C12)⋊21C2, D4.1(C2×C12), C2.13(D4×C12), C6.115(C4×D4), C2.4(C6×SD16), Q8⋊C4⋊21C6, (D4×C12).19C2, D4⋊C4.8C6, (C2×C12).361D4, C42.71(C2×C6), (C2×SD16).5C6, C6.84(C2×SD16), C22.52(C6×D4), C6.117(C4○D8), C4.10(C22×C12), (C6×SD16).10C2, C12.257(C4○D4), (C4×C12).356C22, (C2×C24).437C22, (C2×C12).905C23, C12.155(C22×C4), (C6×D4).291C22, (C6×Q8).254C22, C2.4(C3×C4○D8), C4.2(C3×C4○D4), C4⋊C4.46(C2×C6), (C2×C8).66(C2×C6), (C3×Q8)⋊13(C2×C4), (C3×C4.Q8)⋊29C2, (C2×C4).51(C3×D4), (C3×D4).18(C2×C4), (C2×D4).49(C2×C6), (C2×C6).628(C2×D4), (C2×Q8).51(C2×C6), (C3×Q8⋊C4)⋊44C2, (C2×C4).80(C22×C6), (C3×D4⋊C4).17C2, (C3×C4⋊C4).367C22, SmallGroup(192,871)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×SD16
G = < a,b,c | a12=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 202 in 122 conjugacy classes, 74 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C22×C12, C6×D4, C6×Q8, C4×SD16, C4×C24, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.Q8, D4×C12, Q8×C12, C6×SD16, C12×SD16
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, SD16, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C2×SD16, C4○D8, C3×SD16, C22×C12, C6×D4, C3×C4○D4, C4×SD16, D4×C12, C6×SD16, C3×C4○D8, C12×SD16
►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 59 17 70 42 85 76 26)(2 60 18 71 43 86 77 27)(3 49 19 72 44 87 78 28)(4 50 20 61 45 88 79 29)(5 51 21 62 46 89 80 30)(6 52 22 63 47 90 81 31)(7 53 23 64 48 91 82 32)(8 54 24 65 37 92 83 33)(9 55 13 66 38 93 84 34)(10 56 14 67 39 94 73 35)(11 57 15 68 40 95 74 36)(12 58 16 69 41 96 75 25)
(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 96)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
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), (1,59,17,70,42,85,76,26)(2,60,18,71,43,86,77,27)(3,49,19,72,44,87,78,28)(4,50,20,61,45,88,79,29)(5,51,21,62,46,89,80,30)(6,52,22,63,47,90,81,31)(7,53,23,64,48,91,82,32)(8,54,24,65,37,92,83,33)(9,55,13,66,38,93,84,34)(10,56,14,67,39,94,73,35)(11,57,15,68,40,95,74,36)(12,58,16,69,41,96,75,25), (13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)>;
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), (1,59,17,70,42,85,76,26)(2,60,18,71,43,86,77,27)(3,49,19,72,44,87,78,28)(4,50,20,61,45,88,79,29)(5,51,21,62,46,89,80,30)(6,52,22,63,47,90,81,31)(7,53,23,64,48,91,82,32)(8,54,24,65,37,92,83,33)(9,55,13,66,38,93,84,34)(10,56,14,67,39,94,73,35)(11,57,15,68,40,95,74,36)(12,58,16,69,41,96,75,25), (13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,96)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71) );
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)], [(1,59,17,70,42,85,76,26),(2,60,18,71,43,86,77,27),(3,49,19,72,44,87,78,28),(4,50,20,61,45,88,79,29),(5,51,21,62,46,89,80,30),(6,52,22,63,47,90,81,31),(7,53,23,64,48,91,82,32),(8,54,24,65,37,92,83,33),(9,55,13,66,38,93,84,34),(10,56,14,67,39,94,73,35),(11,57,15,68,40,95,74,36),(12,58,16,69,41,96,75,25)], [(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,96),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(49,72),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AB | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | SD16 | C4○D4 | C3×D4 | C4○D8 | C3×SD16 | C3×C4○D4 | C3×C4○D8 |
| kernel | C12×SD16 | C4×C24 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C4.Q8 | D4×C12 | Q8×C12 | C6×SD16 | C4×SD16 | C3×SD16 | C4×C8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C4×D4 | C4×Q8 | C2×SD16 | SD16 | C2×C12 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C12×SD16 ►in GL3(𝔽73) generated by
| 70 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 72 | 0 | 0 |
| 0 | 0 | 67 |
| 0 | 12 | 12 |
| 72 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [70,0,0,0,27,0,0,0,27],[72,0,0,0,0,12,0,67,12],[72,0,0,0,1,0,0,1,72] >;
C12×SD16 in GAP, Magma, Sage, TeX
C_{12}\times {\rm SD}_{16} % in TeX
G:=Group("C12xSD16"); // GroupNames label
G:=SmallGroup(192,871);
// by ID
G=gap.SmallGroup(192,871);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,268,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations