direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×SD16⋊C4, SD16⋊1C12, C8⋊2(C2×C12), (C4×Q8)⋊6C6, C8⋊C4⋊1C6, C24⋊17(C2×C4), Q8⋊3(C2×C12), C2.D8⋊11C6, (C4×D4).5C6, (Q8×C12)⋊22C2, (C3×SD16)⋊5C4, D4.2(C2×C12), C6.117(C4×D4), C2.15(D4×C12), C42.8(C2×C6), Q8⋊C4⋊16C6, (D4×C12).20C2, D4⋊C4.6C6, (C2×C12).456D4, (C6×SD16).4C2, (C2×SD16).1C6, C22.54(C6×D4), C4.12(C22×C12), C12.259(C4○D4), C6.129(C8⋊C22), (C4×C12).249C22, (C2×C12).907C23, C12.157(C22×C4), (C2×C24).329C22, (C6×D4).292C22, (C6×Q8).256C22, C6.129(C8.C22), (C3×C8⋊C4)⋊6C2, C4.4(C3×C4○D4), C4⋊C4.48(C2×C6), (C2×C8).18(C2×C6), (C3×Q8)⋊14(C2×C4), (C3×C2.D8)⋊26C2, C2.4(C3×C8⋊C22), (C3×D4).19(C2×C4), (C2×D4).50(C2×C6), (C2×C4).102(C3×D4), (C2×C6).630(C2×D4), (C2×Q8).53(C2×C6), C2.4(C3×C8.C22), (C3×Q8⋊C4)⋊39C2, (C2×C4).82(C22×C6), (C3×D4⋊C4).15C2, (C3×C4⋊C4).369C22, SmallGroup(192,873)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×SD16⋊C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >
Subgroups: 202 in 120 conjugacy classes, 70 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, 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, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, 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, SD16⋊C4, C3×C8⋊C4, C3×D4⋊C4, C3×Q8⋊C4, C3×C2.D8, D4×C12, Q8×C12, C6×SD16, C3×SD16⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C8⋊C22, C8.C22, C22×C12, C6×D4, C3×C4○D4, SD16⋊C4, D4×C12, C3×C8⋊C22, C3×C8.C22, C3×SD16⋊C4
►On 96 points
(1 74 91)(2 75 92)(3 76 93)(4 77 94)(5 78 95)(6 79 96)(7 80 89)(8 73 90)(9 68 84)(10 69 85)(11 70 86)(12 71 87)(13 72 88)(14 65 81)(15 66 82)(16 67 83)(17 28 54)(18 29 55)(19 30 56)(20 31 49)(21 32 50)(22 25 51)(23 26 52)(24 27 53)(33 42 59)(34 43 60)(35 44 61)(36 45 62)(37 46 63)(38 47 64)(39 48 57)(40 41 58)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(41 43)(42 46)(45 47)(49 53)(50 56)(52 54)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)(73 79)(75 77)(76 80)(81 85)(82 88)(84 86)(89 93)(90 96)(92 94)
(1 67 55 61)(2 72 56 58)(3 69 49 63)(4 66 50 60)(5 71 51 57)(6 68 52 62)(7 65 53 59)(8 70 54 64)(9 26 45 96)(10 31 46 93)(11 28 47 90)(12 25 48 95)(13 30 41 92)(14 27 42 89)(15 32 43 94)(16 29 44 91)(17 38 73 86)(18 35 74 83)(19 40 75 88)(20 37 76 85)(21 34 77 82)(22 39 78 87)(23 36 79 84)(24 33 80 81)
G:=sub<Sym(96)| (1,74,91)(2,75,92)(3,76,93)(4,77,94)(5,78,95)(6,79,96)(7,80,89)(8,73,90)(9,68,84)(10,69,85)(11,70,86)(12,71,87)(13,72,88)(14,65,81)(15,66,82)(16,67,83)(17,28,54)(18,29,55)(19,30,56)(20,31,49)(21,32,50)(22,25,51)(23,26,52)(24,27,53)(33,42,59)(34,43,60)(35,44,61)(36,45,62)(37,46,63)(38,47,64)(39,48,57)(40,41,58), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(41,43)(42,46)(45,47)(49,53)(50,56)(52,54)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,79)(75,77)(76,80)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94), (1,67,55,61)(2,72,56,58)(3,69,49,63)(4,66,50,60)(5,71,51,57)(6,68,52,62)(7,65,53,59)(8,70,54,64)(9,26,45,96)(10,31,46,93)(11,28,47,90)(12,25,48,95)(13,30,41,92)(14,27,42,89)(15,32,43,94)(16,29,44,91)(17,38,73,86)(18,35,74,83)(19,40,75,88)(20,37,76,85)(21,34,77,82)(22,39,78,87)(23,36,79,84)(24,33,80,81)>;
G:=Group( (1,74,91)(2,75,92)(3,76,93)(4,77,94)(5,78,95)(6,79,96)(7,80,89)(8,73,90)(9,68,84)(10,69,85)(11,70,86)(12,71,87)(13,72,88)(14,65,81)(15,66,82)(16,67,83)(17,28,54)(18,29,55)(19,30,56)(20,31,49)(21,32,50)(22,25,51)(23,26,52)(24,27,53)(33,42,59)(34,43,60)(35,44,61)(36,45,62)(37,46,63)(38,47,64)(39,48,57)(40,41,58), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(41,43)(42,46)(45,47)(49,53)(50,56)(52,54)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,79)(75,77)(76,80)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94), (1,67,55,61)(2,72,56,58)(3,69,49,63)(4,66,50,60)(5,71,51,57)(6,68,52,62)(7,65,53,59)(8,70,54,64)(9,26,45,96)(10,31,46,93)(11,28,47,90)(12,25,48,95)(13,30,41,92)(14,27,42,89)(15,32,43,94)(16,29,44,91)(17,38,73,86)(18,35,74,83)(19,40,75,88)(20,37,76,85)(21,34,77,82)(22,39,78,87)(23,36,79,84)(24,33,80,81) );
G=PermutationGroup([[(1,74,91),(2,75,92),(3,76,93),(4,77,94),(5,78,95),(6,79,96),(7,80,89),(8,73,90),(9,68,84),(10,69,85),(11,70,86),(12,71,87),(13,72,88),(14,65,81),(15,66,82),(16,67,83),(17,28,54),(18,29,55),(19,30,56),(20,31,49),(21,32,50),(22,25,51),(23,26,52),(24,27,53),(33,42,59),(34,43,60),(35,44,61),(36,45,62),(37,46,63),(38,47,64),(39,48,57),(40,41,58)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38),(41,43),(42,46),(45,47),(49,53),(50,56),(52,54),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70),(73,79),(75,77),(76,80),(81,85),(82,88),(84,86),(89,93),(90,96),(92,94)], [(1,67,55,61),(2,72,56,58),(3,69,49,63),(4,66,50,60),(5,71,51,57),(6,68,52,62),(7,65,53,59),(8,70,54,64),(9,26,45,96),(10,31,46,93),(11,28,47,90),(12,25,48,95),(13,30,41,92),(14,27,42,89),(15,32,43,94),(16,29,44,91),(17,38,73,86),(18,35,74,83),(19,40,75,88),(20,37,76,85),(21,34,77,82),(22,39,78,87),(23,36,79,84),(24,33,80,81)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | ··· | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12L | 12M | ··· | 12X | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | C8⋊C22 | C8.C22 | C3×C8⋊C22 | C3×C8.C22 |
| kernel | C3×SD16⋊C4 | C3×C8⋊C4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C2.D8 | D4×C12 | Q8×C12 | C6×SD16 | SD16⋊C4 | C3×SD16 | C8⋊C4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4×D4 | C4×Q8 | C2×SD16 | SD16 | C2×C12 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×SD16⋊C4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 72 | 46 | 0 | 0 | 0 | 0 |
| 19 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 9 | 64 | 8 |
| 0 | 0 | 64 | 64 | 8 | 0 |
| 0 | 0 | 72 | 0 | 17 | 9 |
| 0 | 0 | 1 | 1 | 56 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 54 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 72 | 0 | 72 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[72,19,0,0,0,0,46,1,0,0,0,0,0,0,65,64,72,1,0,0,9,64,0,1,0,0,64,8,17,56,0,0,8,0,9,0],[1,54,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,72,0,1,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,72,0,0,0,1,72,72,2,0,0,1,0,0,0,0,0,1,72,0,1] >;
C3×SD16⋊C4 in GAP, Magma, Sage, TeX
C_3\times {\rm SD}_{16}\rtimes C_4 % in TeX
G:=Group("C3xSD16:C4"); // GroupNames label
G:=SmallGroup(192,873);
// by ID
G=gap.SmallGroup(192,873);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,268,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations