direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C4○D16, D16⋊3C6, Q32⋊3C6, SD32⋊3C6, C12.71D8, C24.76D4, C48.23C22, C24.66C23, (C2×C16)⋊6C6, C4○D8⋊1C6, (C2×C48)⋊13C2, (C3×D16)⋊7C2, C16.6(C2×C6), (C3×Q32)⋊7C2, D8.2(C2×C6), C4.20(C3×D8), (C2×C6).12D8, C4.10(C6×D4), C6.87(C2×D8), C2.15(C6×D8), C8.13(C3×D4), (C3×SD32)⋊7C2, C8.6(C22×C6), Q16.2(C2×C6), C22.1(C3×D8), C12.317(C2×D4), (C2×C12).429D4, (C3×D8).12C22, (C2×C24).412C22, (C3×Q16).14C22, (C3×C4○D8)⋊8C2, (C2×C8).91(C2×C6), (C2×C4).85(C3×D4), SmallGroup(192,941)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4○D16
G = < a,b,c,d | a3=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >
Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C2×C16, D16, SD32, Q32, C4○D8, C48, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C4○D16, C2×C48, C3×D16, C3×SD32, C3×Q32, C3×C4○D8, C3×C4○D16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, C4○D16, C6×D8, C3×C4○D16
►On 96 points
Generators in S96
(1 24 72)(2 25 73)(3 26 74)(4 27 75)(5 28 76)(6 29 77)(7 30 78)(8 31 79)(9 32 80)(10 17 65)(11 18 66)(12 19 67)(13 20 68)(14 21 69)(15 22 70)(16 23 71)(33 50 85)(34 51 86)(35 52 87)(36 53 88)(37 54 89)(38 55 90)(39 56 91)(40 57 92)(41 58 93)(42 59 94)(43 60 95)(44 61 96)(45 62 81)(46 63 82)(47 64 83)(48 49 84)
(1 33 9 41)(2 34 10 42)(3 35 11 43)(4 36 12 44)(5 37 13 45)(6 38 14 46)(7 39 15 47)(8 40 16 48)(17 59 25 51)(18 60 26 52)(19 61 27 53)(20 62 28 54)(21 63 29 55)(22 64 30 56)(23 49 31 57)(24 50 32 58)(65 94 73 86)(66 95 74 87)(67 96 75 88)(68 81 76 89)(69 82 77 90)(70 83 78 91)(71 84 79 92)(72 85 80 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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 50)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)
G:=sub<Sym(96)| (1,24,72)(2,25,73)(3,26,74)(4,27,75)(5,28,76)(6,29,77)(7,30,78)(8,31,79)(9,32,80)(10,17,65)(11,18,66)(12,19,67)(13,20,68)(14,21,69)(15,22,70)(16,23,71)(33,50,85)(34,51,86)(35,52,87)(36,53,88)(37,54,89)(38,55,90)(39,56,91)(40,57,92)(41,58,93)(42,59,94)(43,60,95)(44,61,96)(45,62,81)(46,63,82)(47,64,83)(48,49,84), (1,33,9,41)(2,34,10,42)(3,35,11,43)(4,36,12,44)(5,37,13,45)(6,38,14,46)(7,39,15,47)(8,40,16,48)(17,59,25,51)(18,60,26,52)(19,61,27,53)(20,62,28,54)(21,63,29,55)(22,64,30,56)(23,49,31,57)(24,50,32,58)(65,94,73,86)(66,95,74,87)(67,96,75,88)(68,81,76,89)(69,82,77,90)(70,83,78,91)(71,84,79,92)(72,85,80,93), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,32)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,50)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,88)(82,87)(83,86)(84,85)(89,96)(90,95)(91,94)(92,93)>;
G:=Group( (1,24,72)(2,25,73)(3,26,74)(4,27,75)(5,28,76)(6,29,77)(7,30,78)(8,31,79)(9,32,80)(10,17,65)(11,18,66)(12,19,67)(13,20,68)(14,21,69)(15,22,70)(16,23,71)(33,50,85)(34,51,86)(35,52,87)(36,53,88)(37,54,89)(38,55,90)(39,56,91)(40,57,92)(41,58,93)(42,59,94)(43,60,95)(44,61,96)(45,62,81)(46,63,82)(47,64,83)(48,49,84), (1,33,9,41)(2,34,10,42)(3,35,11,43)(4,36,12,44)(5,37,13,45)(6,38,14,46)(7,39,15,47)(8,40,16,48)(17,59,25,51)(18,60,26,52)(19,61,27,53)(20,62,28,54)(21,63,29,55)(22,64,30,56)(23,49,31,57)(24,50,32,58)(65,94,73,86)(66,95,74,87)(67,96,75,88)(68,81,76,89)(69,82,77,90)(70,83,78,91)(71,84,79,92)(72,85,80,93), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,32)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,50)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,80)(81,88)(82,87)(83,86)(84,85)(89,96)(90,95)(91,94)(92,93) );
G=PermutationGroup([[(1,24,72),(2,25,73),(3,26,74),(4,27,75),(5,28,76),(6,29,77),(7,30,78),(8,31,79),(9,32,80),(10,17,65),(11,18,66),(12,19,67),(13,20,68),(14,21,69),(15,22,70),(16,23,71),(33,50,85),(34,51,86),(35,52,87),(36,53,88),(37,54,89),(38,55,90),(39,56,91),(40,57,92),(41,58,93),(42,59,94),(43,60,95),(44,61,96),(45,62,81),(46,63,82),(47,64,83),(48,49,84)], [(1,33,9,41),(2,34,10,42),(3,35,11,43),(4,36,12,44),(5,37,13,45),(6,38,14,46),(7,39,15,47),(8,40,16,48),(17,59,25,51),(18,60,26,52),(19,61,27,53),(20,62,28,54),(21,63,29,55),(22,64,30,56),(23,49,31,57),(24,50,32,58),(65,94,73,86),(66,95,74,87),(67,96,75,88),(68,81,76,89),(69,82,77,90),(70,83,78,91),(71,84,79,92),(72,85,80,93)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(31,32),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,50),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,80),(81,88),(82,87),(83,86),(84,85),(89,96),(90,95),(91,94),(92,93)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D8 | D8 | C3×D4 | C3×D4 | C3×D8 | C3×D8 | C4○D16 | C3×C4○D16 |
| kernel | C3×C4○D16 | C2×C48 | C3×D16 | C3×SD32 | C3×Q32 | C3×C4○D8 | C4○D16 | C2×C16 | D16 | SD32 | Q32 | C4○D8 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3×C4○D16 ►in GL2(𝔽97) generated by
| 35 | 0 |
| 0 | 35 |
| 75 | 0 |
| 0 | 75 |
| 2 | 26 |
| 71 | 2 |
| 71 | 2 |
| 2 | 26 |
G:=sub<GL(2,GF(97))| [35,0,0,35],[75,0,0,75],[2,71,26,2],[71,2,2,26] >;
C3×C4○D16 in GAP, Magma, Sage, TeX
C_3\times C_4\circ D_{16} % in TeX
G:=Group("C3xC4oD16"); // GroupNames label
G:=SmallGroup(192,941);
// by ID
G=gap.SmallGroup(192,941);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,520,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^8=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^7>;
// generators/relations