direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.4D8, C12.42D8, C12.32SD16, (C4×C8)⋊5C6, C4⋊Q8⋊5C6, C4.4(C3×D8), C2.9(C6×D8), (C4×C24)⋊10C2, C6.81(C2×D8), D4⋊C4⋊3C6, C4⋊1D4.4C6, C4.3(C3×SD16), (C2×C12).419D4, C42.76(C2×C6), C2.14(C6×SD16), C6.94(C2×SD16), C22.107(C6×D4), C12.268(C4○D4), (C4×C12).360C22, (C2×C12).942C23, (C2×C24).366C22, C6.71(C4.4D4), (C6×D4).198C22, (C3×C4⋊Q8)⋊26C2, C4⋊C4.17(C2×C6), (C2×C8).68(C2×C6), C4.13(C3×C4○D4), (C2×C4).75(C3×D4), (C3×D4⋊C4)⋊3C2, (C2×D4).21(C2×C6), (C2×C6).663(C2×D4), C2.9(C3×C4.4D4), (C3×C4⋊1D4).11C2, (C3×C4⋊C4).237C22, (C2×C4).117(C22×C6), SmallGroup(192,919)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.4D8
G = < a,b,c,d | a3=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 258 in 118 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×D4, C6×D4, C6×Q8, C4.4D8, C4×C24, C3×D4⋊C4, C3×C4⋊1D4, C3×C4⋊Q8, C3×C4.4D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, SD16, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C2×D8, C2×SD16, C3×D8, C3×SD16, C6×D4, C3×C4○D4, C4.4D8, C3×C4.4D4, C6×D8, C6×SD16, C3×C4.4D8
►On 96 points
(1 22 57)(2 23 58)(3 24 59)(4 17 60)(5 18 61)(6 19 62)(7 20 63)(8 21 64)(9 25 68)(10 26 69)(11 27 70)(12 28 71)(13 29 72)(14 30 65)(15 31 66)(16 32 67)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(41 53 92)(42 54 93)(43 55 94)(44 56 95)(45 49 96)(46 50 89)(47 51 90)(48 52 91)
(1 41 75 66)(2 42 76 67)(3 43 77 68)(4 44 78 69)(5 45 79 70)(6 46 80 71)(7 47 73 72)(8 48 74 65)(9 24 55 85)(10 17 56 86)(11 18 49 87)(12 19 50 88)(13 20 51 81)(14 21 52 82)(15 22 53 83)(16 23 54 84)(25 59 94 34)(26 60 95 35)(27 61 96 36)(28 62 89 37)(29 63 90 38)(30 64 91 39)(31 57 92 40)(32 58 93 33)
(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 48 75 65)(2 72 76 47)(3 46 77 71)(4 70 78 45)(5 44 79 69)(6 68 80 43)(7 42 73 67)(8 66 74 41)(9 88 55 19)(10 18 56 87)(11 86 49 17)(12 24 50 85)(13 84 51 23)(14 22 52 83)(15 82 53 21)(16 20 54 81)(25 37 94 62)(26 61 95 36)(27 35 96 60)(28 59 89 34)(29 33 90 58)(30 57 91 40)(31 39 92 64)(32 63 93 38)
G:=sub<Sym(96)| (1,22,57)(2,23,58)(3,24,59)(4,17,60)(5,18,61)(6,19,62)(7,20,63)(8,21,64)(9,25,68)(10,26,69)(11,27,70)(12,28,71)(13,29,72)(14,30,65)(15,31,66)(16,32,67)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(41,53,92)(42,54,93)(43,55,94)(44,56,95)(45,49,96)(46,50,89)(47,51,90)(48,52,91), (1,41,75,66)(2,42,76,67)(3,43,77,68)(4,44,78,69)(5,45,79,70)(6,46,80,71)(7,47,73,72)(8,48,74,65)(9,24,55,85)(10,17,56,86)(11,18,49,87)(12,19,50,88)(13,20,51,81)(14,21,52,82)(15,22,53,83)(16,23,54,84)(25,59,94,34)(26,60,95,35)(27,61,96,36)(28,62,89,37)(29,63,90,38)(30,64,91,39)(31,57,92,40)(32,58,93,33), (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), (1,48,75,65)(2,72,76,47)(3,46,77,71)(4,70,78,45)(5,44,79,69)(6,68,80,43)(7,42,73,67)(8,66,74,41)(9,88,55,19)(10,18,56,87)(11,86,49,17)(12,24,50,85)(13,84,51,23)(14,22,52,83)(15,82,53,21)(16,20,54,81)(25,37,94,62)(26,61,95,36)(27,35,96,60)(28,59,89,34)(29,33,90,58)(30,57,91,40)(31,39,92,64)(32,63,93,38)>;
G:=Group( (1,22,57)(2,23,58)(3,24,59)(4,17,60)(5,18,61)(6,19,62)(7,20,63)(8,21,64)(9,25,68)(10,26,69)(11,27,70)(12,28,71)(13,29,72)(14,30,65)(15,31,66)(16,32,67)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(41,53,92)(42,54,93)(43,55,94)(44,56,95)(45,49,96)(46,50,89)(47,51,90)(48,52,91), (1,41,75,66)(2,42,76,67)(3,43,77,68)(4,44,78,69)(5,45,79,70)(6,46,80,71)(7,47,73,72)(8,48,74,65)(9,24,55,85)(10,17,56,86)(11,18,49,87)(12,19,50,88)(13,20,51,81)(14,21,52,82)(15,22,53,83)(16,23,54,84)(25,59,94,34)(26,60,95,35)(27,61,96,36)(28,62,89,37)(29,63,90,38)(30,64,91,39)(31,57,92,40)(32,58,93,33), (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), (1,48,75,65)(2,72,76,47)(3,46,77,71)(4,70,78,45)(5,44,79,69)(6,68,80,43)(7,42,73,67)(8,66,74,41)(9,88,55,19)(10,18,56,87)(11,86,49,17)(12,24,50,85)(13,84,51,23)(14,22,52,83)(15,82,53,21)(16,20,54,81)(25,37,94,62)(26,61,95,36)(27,35,96,60)(28,59,89,34)(29,33,90,58)(30,57,91,40)(31,39,92,64)(32,63,93,38) );
G=PermutationGroup([[(1,22,57),(2,23,58),(3,24,59),(4,17,60),(5,18,61),(6,19,62),(7,20,63),(8,21,64),(9,25,68),(10,26,69),(11,27,70),(12,28,71),(13,29,72),(14,30,65),(15,31,66),(16,32,67),(33,76,84),(34,77,85),(35,78,86),(36,79,87),(37,80,88),(38,73,81),(39,74,82),(40,75,83),(41,53,92),(42,54,93),(43,55,94),(44,56,95),(45,49,96),(46,50,89),(47,51,90),(48,52,91)], [(1,41,75,66),(2,42,76,67),(3,43,77,68),(4,44,78,69),(5,45,79,70),(6,46,80,71),(7,47,73,72),(8,48,74,65),(9,24,55,85),(10,17,56,86),(11,18,49,87),(12,19,50,88),(13,20,51,81),(14,21,52,82),(15,22,53,83),(16,23,54,84),(25,59,94,34),(26,60,95,35),(27,61,96,36),(28,62,89,37),(29,63,90,38),(30,64,91,39),(31,57,92,40),(32,58,93,33)], [(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)], [(1,48,75,65),(2,72,76,47),(3,46,77,71),(4,70,78,45),(5,44,79,69),(6,68,80,43),(7,42,73,67),(8,66,74,41),(9,88,55,19),(10,18,56,87),(11,86,49,17),(12,24,50,85),(13,84,51,23),(14,22,52,83),(15,82,53,21),(16,20,54,81),(25,37,94,62),(26,61,95,36),(27,35,96,60),(28,59,89,34),(29,33,90,58),(30,57,91,40),(31,39,92,64),(32,63,93,38)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12L | 12M | 12N | 12O | 12P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 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 | 8 | 8 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 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 | C3 | C6 | C6 | C6 | C6 | D4 | D8 | SD16 | C4○D4 | C3×D4 | C3×D8 | C3×SD16 | C3×C4○D4 |
| kernel | C3×C4.4D8 | C4×C24 | C3×D4⋊C4 | C3×C4⋊1D4 | C3×C4⋊Q8 | C4.4D8 | C4×C8 | D4⋊C4 | C4⋊1D4 | C4⋊Q8 | C2×C12 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C4.4D8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 27 | 19 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 1 | 71 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 67 | 6 |
| 27 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 67 | 67 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[27,0,0,0,19,46,0,0,0,0,0,1,0,0,72,0],[1,0,0,0,71,72,0,0,0,0,6,67,0,0,6,6],[27,27,0,0,0,46,0,0,0,0,6,67,0,0,67,67] >;
C3×C4.4D8 in GAP, Magma, Sage, TeX
C_3\times C_4._4D_8
% in TeX
G:=Group("C3xC4.4D8"); // GroupNames label
G:=SmallGroup(192,919);
// by ID
G=gap.SmallGroup(192,919);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,344,1094,142,4204,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations