direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×D4○C16, D4.2C24, Q8.3C24, M5(2)⋊7C6, C48.29C22, C24.83C23, M4(2).4C12, (C2×C16)⋊9C6, (C2×C48)⋊19C2, C16.8(C2×C6), C4.5(C2×C24), C8○D4.5C6, (C3×D4).4C8, (C3×Q8).4C8, C12.34(C2×C8), C24.68(C2×C4), C8.12(C2×C12), C4○D4.6C12, C2.7(C22×C24), C22.1(C2×C24), C8.16(C22×C6), C6.36(C22×C8), (C3×M5(2))⋊15C2, C4.36(C22×C12), (C3×M4(2)).8C4, (C2×C24).449C22, C12.194(C22×C4), (C2×C6).8(C2×C8), (C3×C4○D4).7C4, (C3×C8○D4).6C2, (C2×C4).51(C2×C12), (C2×C8).103(C2×C6), (C2×C12).272(C2×C4), SmallGroup(192,937)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4○C16
G = < a,b,c,d | a3=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C16, C16, C2×C8, M4(2), C4○D4, C24, C24, C2×C12, C3×D4, C3×Q8, C2×C16, M5(2), C8○D4, C48, C48, C2×C24, C3×M4(2), C3×C4○D4, D4○C16, C2×C48, C3×M5(2), C3×C8○D4, C3×D4○C16
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, C22×C4, C24, C2×C12, C22×C6, C22×C8, C2×C24, C22×C12, D4○C16, C22×C24, C3×D4○C16
►On 96 points
Generators in S96
(1 89 61)(2 90 62)(3 91 63)(4 92 64)(5 93 49)(6 94 50)(7 95 51)(8 96 52)(9 81 53)(10 82 54)(11 83 55)(12 84 56)(13 85 57)(14 86 58)(15 87 59)(16 88 60)(17 65 34)(18 66 35)(19 67 36)(20 68 37)(21 69 38)(22 70 39)(23 71 40)(24 72 41)(25 73 42)(26 74 43)(27 75 44)(28 76 45)(29 77 46)(30 78 47)(31 79 48)(32 80 33)
(1 78 9 70)(2 79 10 71)(3 80 11 72)(4 65 12 73)(5 66 13 74)(6 67 14 75)(7 68 15 76)(8 69 16 77)(17 56 25 64)(18 57 26 49)(19 58 27 50)(20 59 28 51)(21 60 29 52)(22 61 30 53)(23 62 31 54)(24 63 32 55)(33 83 41 91)(34 84 42 92)(35 85 43 93)(36 86 44 94)(37 87 45 95)(38 88 46 96)(39 89 47 81)(40 90 48 82)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 65)(13 66)(14 67)(15 68)(16 69)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 81)(48 82)
(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)
G:=sub<Sym(96)| (1,89,61)(2,90,62)(3,91,63)(4,92,64)(5,93,49)(6,94,50)(7,95,51)(8,96,52)(9,81,53)(10,82,54)(11,83,55)(12,84,56)(13,85,57)(14,86,58)(15,87,59)(16,88,60)(17,65,34)(18,66,35)(19,67,36)(20,68,37)(21,69,38)(22,70,39)(23,71,40)(24,72,41)(25,73,42)(26,74,43)(27,75,44)(28,76,45)(29,77,46)(30,78,47)(31,79,48)(32,80,33), (1,78,9,70)(2,79,10,71)(3,80,11,72)(4,65,12,73)(5,66,13,74)(6,67,14,75)(7,68,15,76)(8,69,16,77)(17,56,25,64)(18,57,26,49)(19,58,27,50)(20,59,28,51)(21,60,29,52)(22,61,30,53)(23,62,31,54)(24,63,32,55)(33,83,41,91)(34,84,42,92)(35,85,43,93)(36,86,44,94)(37,87,45,95)(38,88,46,96)(39,89,47,81)(40,90,48,82), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,65)(13,66)(14,67)(15,68)(16,69)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,81)(48,82), (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)>;
G:=Group( (1,89,61)(2,90,62)(3,91,63)(4,92,64)(5,93,49)(6,94,50)(7,95,51)(8,96,52)(9,81,53)(10,82,54)(11,83,55)(12,84,56)(13,85,57)(14,86,58)(15,87,59)(16,88,60)(17,65,34)(18,66,35)(19,67,36)(20,68,37)(21,69,38)(22,70,39)(23,71,40)(24,72,41)(25,73,42)(26,74,43)(27,75,44)(28,76,45)(29,77,46)(30,78,47)(31,79,48)(32,80,33), (1,78,9,70)(2,79,10,71)(3,80,11,72)(4,65,12,73)(5,66,13,74)(6,67,14,75)(7,68,15,76)(8,69,16,77)(17,56,25,64)(18,57,26,49)(19,58,27,50)(20,59,28,51)(21,60,29,52)(22,61,30,53)(23,62,31,54)(24,63,32,55)(33,83,41,91)(34,84,42,92)(35,85,43,93)(36,86,44,94)(37,87,45,95)(38,88,46,96)(39,89,47,81)(40,90,48,82), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,65)(13,66)(14,67)(15,68)(16,69)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,81)(48,82), (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) );
G=PermutationGroup([[(1,89,61),(2,90,62),(3,91,63),(4,92,64),(5,93,49),(6,94,50),(7,95,51),(8,96,52),(9,81,53),(10,82,54),(11,83,55),(12,84,56),(13,85,57),(14,86,58),(15,87,59),(16,88,60),(17,65,34),(18,66,35),(19,67,36),(20,68,37),(21,69,38),(22,70,39),(23,71,40),(24,72,41),(25,73,42),(26,74,43),(27,75,44),(28,76,45),(29,77,46),(30,78,47),(31,79,48),(32,80,33)], [(1,78,9,70),(2,79,10,71),(3,80,11,72),(4,65,12,73),(5,66,13,74),(6,67,14,75),(7,68,15,76),(8,69,16,77),(17,56,25,64),(18,57,26,49),(19,58,27,50),(20,59,28,51),(21,60,29,52),(22,61,30,53),(23,62,31,54),(24,63,32,55),(33,83,41,91),(34,84,42,92),(35,85,43,93),(36,86,44,94),(37,87,45,95),(38,88,46,96),(39,89,47,81),(40,90,48,82)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,65),(13,66),(14,67),(15,68),(16,69),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,81),(48,82)], [(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)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | ··· | 16H | 16I | ··· | 16T | 24A | ··· | 24H | 24I | ··· | 24T | 48A | ··· | 48P | 48Q | ··· | 48AN |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 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 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | D4○C16 | C3×D4○C16 |
| kernel | C3×D4○C16 | C2×C48 | C3×M5(2) | C3×C8○D4 | D4○C16 | C3×M4(2) | C3×C4○D4 | C2×C16 | M5(2) | C8○D4 | C3×D4 | C3×Q8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 2 | 6 | 6 | 2 | 12 | 4 | 12 | 4 | 24 | 8 | 8 | 16 |
Matrix representation of C3×D4○C16 ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 79 | 80 |
| 0 | 2 | 18 |
| 96 | 0 | 0 |
| 0 | 18 | 16 |
| 0 | 95 | 79 |
| 96 | 0 | 0 |
| 0 | 70 | 0 |
| 0 | 0 | 70 |
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,79,2,0,80,18],[96,0,0,0,18,95,0,16,79],[96,0,0,0,70,0,0,0,70] >;
C3×D4○C16 in GAP, Magma, Sage, TeX
C_3\times D_4\circ C_{16} % in TeX
G:=Group("C3xD4oC16"); // GroupNames label
G:=SmallGroup(192,937);
// by ID
G=gap.SmallGroup(192,937);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,1059,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations