metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊1C8, C12.7Q8, C4.16D12, C12.32D4, C42.2S3, C4.7Dic6, C6.5M4(2), C4⋊(C3⋊C8), C3⋊1(C4⋊C8), C6.7(C2×C8), C6.1(C4⋊C4), (C2×C12).7C4, (C4×C12).4C2, (C2×C4).89D6, (C2×C4).3Dic3, C2.1(C4⋊Dic3), C2.2(C4.Dic3), C22.8(C2×Dic3), (C2×C12).103C22, C2.3(C2×C3⋊C8), (C2×C3⋊C8).8C2, (C2×C6).26(C2×C4), SmallGroup(96,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊C8
G = < a,b | a12=b8=1, bab-1=a-1 >
Smallest permutation representation of C12⋊C8
►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 83 47 61 91 14 53 34)(2 82 48 72 92 13 54 33)(3 81 37 71 93 24 55 32)(4 80 38 70 94 23 56 31)(5 79 39 69 95 22 57 30)(6 78 40 68 96 21 58 29)(7 77 41 67 85 20 59 28)(8 76 42 66 86 19 60 27)(9 75 43 65 87 18 49 26)(10 74 44 64 88 17 50 25)(11 73 45 63 89 16 51 36)(12 84 46 62 90 15 52 35)
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), (1,83,47,61,91,14,53,34)(2,82,48,72,92,13,54,33)(3,81,37,71,93,24,55,32)(4,80,38,70,94,23,56,31)(5,79,39,69,95,22,57,30)(6,78,40,68,96,21,58,29)(7,77,41,67,85,20,59,28)(8,76,42,66,86,19,60,27)(9,75,43,65,87,18,49,26)(10,74,44,64,88,17,50,25)(11,73,45,63,89,16,51,36)(12,84,46,62,90,15,52,35)>;
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), (1,83,47,61,91,14,53,34)(2,82,48,72,92,13,54,33)(3,81,37,71,93,24,55,32)(4,80,38,70,94,23,56,31)(5,79,39,69,95,22,57,30)(6,78,40,68,96,21,58,29)(7,77,41,67,85,20,59,28)(8,76,42,66,86,19,60,27)(9,75,43,65,87,18,49,26)(10,74,44,64,88,17,50,25)(11,73,45,63,89,16,51,36)(12,84,46,62,90,15,52,35) );
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)], [(1,83,47,61,91,14,53,34),(2,82,48,72,92,13,54,33),(3,81,37,71,93,24,55,32),(4,80,38,70,94,23,56,31),(5,79,39,69,95,22,57,30),(6,78,40,68,96,21,58,29),(7,77,41,67,85,20,59,28),(8,76,42,66,86,19,60,27),(9,75,43,65,87,18,49,26),(10,74,44,64,88,17,50,25),(11,73,45,63,89,16,51,36),(12,84,46,62,90,15,52,35)]])
C12⋊C8 is a maximal subgroup of
C4.8Dic12 C24⋊2C8 C24⋊1C8 C4.17D24 C12.53D8 C12.39SD16 C4.Dic12 C12.47D8 C4.D24 C12.2D8 C12.57D8 C12.26Q16 C12.9D8 C12.5Q16 C12.10D8 C8×Dic6 C24⋊12Q8 C8×D12 C8⋊6D12 C24⋊Q8 C8⋊9D12 S3×C4⋊C8 C42.200D6 C42.202D6 C12⋊M4(2) C42.30D6 C42.31D6 C12⋊7M4(2) C42.285D6 C42.270D6 C42.43D6 C42.187D6 C12.50D8 C12.38SD16 D4.3Dic6 D4×C3⋊C8 C42.47D6 C12⋊3M4(2) C12⋊7D8 D4.1D12 D4.2D12 Q8⋊4Dic6 Q8⋊5Dic6 Q8.5Dic6 Q8×C3⋊C8 C42.210D6 Q8⋊2D12 Q8.6D12 C12⋊7Q16 C42.61D6 D12.23D4 Dic6.4Q8 D12.4Q8 C12⋊2D8 Dic6⋊9D4 C12⋊5SD16 D12⋊5Q8 D12⋊6Q8 C12⋊Q16 Dic6⋊5Q8 Dic6⋊6Q8 C36⋊C8 C12.81D12 C12.57D12 C60.13Q8 C60⋊5C8 C60⋊C8 Dic5.13D12
C12⋊C8 is a maximal quotient of
C24⋊2C8 C24⋊1C8 C12⋊C16 C24.1C8 (C2×C12)⋊3C8 C36⋊C8 C12.81D12 C12.57D12 C60.13Q8 C60⋊5C8 C60⋊C8 Dic5.13D12
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | Dic3 | D6 | M4(2) | C3⋊C8 | Dic6 | D12 | C4.Dic3 |
| kernel | C12⋊C8 | C2×C3⋊C8 | C4×C12 | C2×C12 | C12 | C42 | C12 | C12 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C12⋊C8 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 54 | 48 | 0 | 0 |
| 0 | 29 | 19 | 0 | 0 |
| 0 | 0 | 0 | 36 | 65 |
| 0 | 0 | 0 | 65 | 37 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0],[10,0,0,0,0,0,54,29,0,0,0,48,19,0,0,0,0,0,36,65,0,0,0,65,37] >;
C12⋊C8 in GAP, Magma, Sage, TeX
C_{12}\rtimes C_8 % in TeX
G:=Group("C12:C8"); // GroupNames label
G:=SmallGroup(96,11);
// by ID
G=gap.SmallGroup(96,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,86,2309]);
// Polycyclic
G:=Group<a,b|a^12=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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