direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×Dic3, C24⋊4C4, C6.3C42, C3⋊C8⋊5C4, C3⋊2(C4×C8), C8○2(C3⋊C8), C2.2(S3×C8), C6.3(C2×C8), C4.20(C4×S3), (C2×C8).10S3, (C2×C4).90D6, C12.40(C2×C4), (C2×C24).14C2, C22.8(C4×S3), C2.2(C4×Dic3), (C2×Dic3).7C4, C4.12(C2×Dic3), (C4×Dic3).10C2, (C2×C12).104C22, C8○(C2×C3⋊C8), (C2×C8)○(C3⋊C8), (C2×C3⋊C8).12C2, (C2×C6).9(C2×C4), (C2×C8)○(C4×Dic3), SmallGroup(96,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C8×Dic3 |
Generators and relations for C8×Dic3
G = < a,b,c | a8=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C8×Dic3
►Regular action 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 30 73 24 86 71)(2 31 74 17 87 72)(3 32 75 18 88 65)(4 25 76 19 81 66)(5 26 77 20 82 67)(6 27 78 21 83 68)(7 28 79 22 84 69)(8 29 80 23 85 70)(9 37 47 58 50 96)(10 38 48 59 51 89)(11 39 41 60 52 90)(12 40 42 61 53 91)(13 33 43 62 54 92)(14 34 44 63 55 93)(15 35 45 64 56 94)(16 36 46 57 49 95)
(1 64 24 15)(2 57 17 16)(3 58 18 9)(4 59 19 10)(5 60 20 11)(6 61 21 12)(7 62 22 13)(8 63 23 14)(25 48 81 89)(26 41 82 90)(27 42 83 91)(28 43 84 92)(29 44 85 93)(30 45 86 94)(31 46 87 95)(32 47 88 96)(33 69 54 79)(34 70 55 80)(35 71 56 73)(36 72 49 74)(37 65 50 75)(38 66 51 76)(39 67 52 77)(40 68 53 78)
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), (1,30,73,24,86,71)(2,31,74,17,87,72)(3,32,75,18,88,65)(4,25,76,19,81,66)(5,26,77,20,82,67)(6,27,78,21,83,68)(7,28,79,22,84,69)(8,29,80,23,85,70)(9,37,47,58,50,96)(10,38,48,59,51,89)(11,39,41,60,52,90)(12,40,42,61,53,91)(13,33,43,62,54,92)(14,34,44,63,55,93)(15,35,45,64,56,94)(16,36,46,57,49,95), (1,64,24,15)(2,57,17,16)(3,58,18,9)(4,59,19,10)(5,60,20,11)(6,61,21,12)(7,62,22,13)(8,63,23,14)(25,48,81,89)(26,41,82,90)(27,42,83,91)(28,43,84,92)(29,44,85,93)(30,45,86,94)(31,46,87,95)(32,47,88,96)(33,69,54,79)(34,70,55,80)(35,71,56,73)(36,72,49,74)(37,65,50,75)(38,66,51,76)(39,67,52,77)(40,68,53,78)>;
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), (1,30,73,24,86,71)(2,31,74,17,87,72)(3,32,75,18,88,65)(4,25,76,19,81,66)(5,26,77,20,82,67)(6,27,78,21,83,68)(7,28,79,22,84,69)(8,29,80,23,85,70)(9,37,47,58,50,96)(10,38,48,59,51,89)(11,39,41,60,52,90)(12,40,42,61,53,91)(13,33,43,62,54,92)(14,34,44,63,55,93)(15,35,45,64,56,94)(16,36,46,57,49,95), (1,64,24,15)(2,57,17,16)(3,58,18,9)(4,59,19,10)(5,60,20,11)(6,61,21,12)(7,62,22,13)(8,63,23,14)(25,48,81,89)(26,41,82,90)(27,42,83,91)(28,43,84,92)(29,44,85,93)(30,45,86,94)(31,46,87,95)(32,47,88,96)(33,69,54,79)(34,70,55,80)(35,71,56,73)(36,72,49,74)(37,65,50,75)(38,66,51,76)(39,67,52,77)(40,68,53,78) );
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)], [(1,30,73,24,86,71),(2,31,74,17,87,72),(3,32,75,18,88,65),(4,25,76,19,81,66),(5,26,77,20,82,67),(6,27,78,21,83,68),(7,28,79,22,84,69),(8,29,80,23,85,70),(9,37,47,58,50,96),(10,38,48,59,51,89),(11,39,41,60,52,90),(12,40,42,61,53,91),(13,33,43,62,54,92),(14,34,44,63,55,93),(15,35,45,64,56,94),(16,36,46,57,49,95)], [(1,64,24,15),(2,57,17,16),(3,58,18,9),(4,59,19,10),(5,60,20,11),(6,61,21,12),(7,62,22,13),(8,63,23,14),(25,48,81,89),(26,41,82,90),(27,42,83,91),(28,43,84,92),(29,44,85,93),(30,45,86,94),(31,46,87,95),(32,47,88,96),(33,69,54,79),(34,70,55,80),(35,71,56,73),(36,72,49,74),(37,65,50,75),(38,66,51,76),(39,67,52,77),(40,68,53,78)]])
C8×Dic3 is a maximal subgroup of
Dic3⋊C16 C48⋊10C4 C24.97D4 S3×C4×C8 D6.C42 C24⋊Q8 Dic3⋊5M4(2) D6.4C42 Dic3.5M4(2) C24⋊C4⋊C2 C3⋊D4⋊C8 Dic3⋊M4(2) Dic3⋊4D8 Dic3⋊6SD16 Dic3.SD16 (C2×C8).200D6 Dic3⋊7SD16 Dic3⋊4Q16 Dic3.1Q16 Q8⋊C4⋊S3 C42.27D6 Dic6⋊C8 C42.200D6 C42.31D6 Dic3⋊8SD16 C24⋊5Q8 C8.8Dic6 Dic3⋊5D8 Dic3⋊5Q16 C24⋊2Q8 C8.6Dic6 D24⋊7C4 C12.12C42 C12.7C42 C24⋊21D4 C24.100D4 C24⋊5D4 C24.22D4 C24.43D4 C24⋊15D4 C24.26D4 C24.28D4 D8⋊5Dic3 C6.(S3×C8) Dic15⋊4C8 C30.C42
C8×Dic3 is a maximal quotient of
C42.279D6 C48⋊10C4 (C2×C24)⋊5C4 C6.(S3×C8) Dic15⋊4C8 C30.C42
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | Dic3 | D6 | C4×S3 | C4×S3 | S3×C8 |
| kernel | C8×Dic3 | C2×C3⋊C8 | C4×Dic3 | C2×C24 | C3⋊C8 | C24 | C2×Dic3 | Dic3 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C8×Dic3 ►in GL3(𝔽73) generated by
| 27 | 0 | 0 |
| 0 | 51 | 0 |
| 0 | 0 | 51 |
| 72 | 0 | 0 |
| 0 | 72 | 1 |
| 0 | 72 | 0 |
| 27 | 0 | 0 |
| 0 | 2 | 60 |
| 0 | 62 | 71 |
G:=sub<GL(3,GF(73))| [27,0,0,0,51,0,0,0,51],[72,0,0,0,72,72,0,1,0],[27,0,0,0,2,62,0,60,71] >;
C8×Dic3 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_3 % in TeX
G:=Group("C8xDic3"); // GroupNames label
G:=SmallGroup(96,20);
// by ID
G=gap.SmallGroup(96,20);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,55,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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