direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.Q8, C8⋊2C12, C24⋊6C4, C12.9Q8, C6.11SD16, C4⋊C4.2C6, (C2×C8).6C6, C4.1(C3×Q8), C4.6(C2×C12), (C2×C6).48D4, C6.12(C4⋊C4), (C2×C24).16C2, C12.43(C2×C4), C2.3(C3×SD16), C22.10(C3×D4), (C2×C12).117C22, C2.3(C3×C4⋊C4), (C3×C4⋊C4).9C2, (C2×C4).20(C2×C6), SmallGroup(96,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.Q8
G = < a,b,c,d | a3=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C3×C4.Q8
►Regular action on 96 points
Generators in S96
(1 38 11)(2 39 12)(3 40 13)(4 33 14)(5 34 15)(6 35 16)(7 36 9)(8 37 10)(17 53 41)(18 54 42)(19 55 43)(20 56 44)(21 49 45)(22 50 46)(23 51 47)(24 52 48)(25 91 71)(26 92 72)(27 93 65)(28 94 66)(29 95 67)(30 96 68)(31 89 69)(32 90 70)(57 74 85)(58 75 86)(59 76 87)(60 77 88)(61 78 81)(62 79 82)(63 80 83)(64 73 84)
(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(25 78 29 74)(26 79 30 75)(27 80 31 76)(28 73 32 77)(33 56 37 52)(34 49 38 53)(35 50 39 54)(36 51 40 55)(57 71 61 67)(58 72 62 68)(59 65 63 69)(60 66 64 70)(81 95 85 91)(82 96 86 92)(83 89 87 93)(84 90 88 94)
(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 65 23 61)(2 68 24 64)(3 71 17 59)(4 66 18 62)(5 69 19 57)(6 72 20 60)(7 67 21 63)(8 70 22 58)(9 95 45 83)(10 90 46 86)(11 93 47 81)(12 96 48 84)(13 91 41 87)(14 94 42 82)(15 89 43 85)(16 92 44 88)(25 53 76 40)(26 56 77 35)(27 51 78 38)(28 54 79 33)(29 49 80 36)(30 52 73 39)(31 55 74 34)(32 50 75 37)
G:=sub<Sym(96)| (1,38,11)(2,39,12)(3,40,13)(4,33,14)(5,34,15)(6,35,16)(7,36,9)(8,37,10)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,91,71)(26,92,72)(27,93,65)(28,94,66)(29,95,67)(30,96,68)(31,89,69)(32,90,70)(57,74,85)(58,75,86)(59,76,87)(60,77,88)(61,78,81)(62,79,82)(63,80,83)(64,73,84), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(25,78,29,74)(26,79,30,75)(27,80,31,76)(28,73,32,77)(33,56,37,52)(34,49,38,53)(35,50,39,54)(36,51,40,55)(57,71,61,67)(58,72,62,68)(59,65,63,69)(60,66,64,70)(81,95,85,91)(82,96,86,92)(83,89,87,93)(84,90,88,94), (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), (1,65,23,61)(2,68,24,64)(3,71,17,59)(4,66,18,62)(5,69,19,57)(6,72,20,60)(7,67,21,63)(8,70,22,58)(9,95,45,83)(10,90,46,86)(11,93,47,81)(12,96,48,84)(13,91,41,87)(14,94,42,82)(15,89,43,85)(16,92,44,88)(25,53,76,40)(26,56,77,35)(27,51,78,38)(28,54,79,33)(29,49,80,36)(30,52,73,39)(31,55,74,34)(32,50,75,37)>;
G:=Group( (1,38,11)(2,39,12)(3,40,13)(4,33,14)(5,34,15)(6,35,16)(7,36,9)(8,37,10)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,91,71)(26,92,72)(27,93,65)(28,94,66)(29,95,67)(30,96,68)(31,89,69)(32,90,70)(57,74,85)(58,75,86)(59,76,87)(60,77,88)(61,78,81)(62,79,82)(63,80,83)(64,73,84), (1,17,5,21)(2,18,6,22)(3,19,7,23)(4,20,8,24)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(25,78,29,74)(26,79,30,75)(27,80,31,76)(28,73,32,77)(33,56,37,52)(34,49,38,53)(35,50,39,54)(36,51,40,55)(57,71,61,67)(58,72,62,68)(59,65,63,69)(60,66,64,70)(81,95,85,91)(82,96,86,92)(83,89,87,93)(84,90,88,94), (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), (1,65,23,61)(2,68,24,64)(3,71,17,59)(4,66,18,62)(5,69,19,57)(6,72,20,60)(7,67,21,63)(8,70,22,58)(9,95,45,83)(10,90,46,86)(11,93,47,81)(12,96,48,84)(13,91,41,87)(14,94,42,82)(15,89,43,85)(16,92,44,88)(25,53,76,40)(26,56,77,35)(27,51,78,38)(28,54,79,33)(29,49,80,36)(30,52,73,39)(31,55,74,34)(32,50,75,37) );
G=PermutationGroup([[(1,38,11),(2,39,12),(3,40,13),(4,33,14),(5,34,15),(6,35,16),(7,36,9),(8,37,10),(17,53,41),(18,54,42),(19,55,43),(20,56,44),(21,49,45),(22,50,46),(23,51,47),(24,52,48),(25,91,71),(26,92,72),(27,93,65),(28,94,66),(29,95,67),(30,96,68),(31,89,69),(32,90,70),(57,74,85),(58,75,86),(59,76,87),(60,77,88),(61,78,81),(62,79,82),(63,80,83),(64,73,84)], [(1,17,5,21),(2,18,6,22),(3,19,7,23),(4,20,8,24),(9,47,13,43),(10,48,14,44),(11,41,15,45),(12,42,16,46),(25,78,29,74),(26,79,30,75),(27,80,31,76),(28,73,32,77),(33,56,37,52),(34,49,38,53),(35,50,39,54),(36,51,40,55),(57,71,61,67),(58,72,62,68),(59,65,63,69),(60,66,64,70),(81,95,85,91),(82,96,86,92),(83,89,87,93),(84,90,88,94)], [(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)], [(1,65,23,61),(2,68,24,64),(3,71,17,59),(4,66,18,62),(5,69,19,57),(6,72,20,60),(7,67,21,63),(8,70,22,58),(9,95,45,83),(10,90,46,86),(11,93,47,81),(12,96,48,84),(13,91,41,87),(14,94,42,82),(15,89,43,85),(16,92,44,88),(25,53,76,40),(26,56,77,35),(27,51,78,38),(28,54,79,33),(29,49,80,36),(30,52,73,39),(31,55,74,34),(32,50,75,37)]])
C3×C4.Q8 is a maximal subgroup of
C8.Dic6 D24⋊8C4 Dic3⋊8SD16 Dic12⋊9C4 Dic6⋊Q8 C24⋊5Q8 C24⋊3Q8 Dic6.Q8 C8.8Dic6 (S3×C8)⋊C4 C8⋊(C4×S3) D6.2SD16 D6.4SD16 C8⋊8D12 C24⋊7D4 C4.Q8⋊S3 C8.2D12 C6.(C4○D8) D24⋊9C4 D12⋊Q8 D12.Q8 C12×SD16
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | SD16 | C3×Q8 | C3×D4 | C3×SD16 |
| kernel | C3×C4.Q8 | C3×C4⋊C4 | C2×C24 | C4.Q8 | C24 | C4⋊C4 | C2×C8 | C8 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C3×C4.Q8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 16 | 53 |
| 0 | 0 | 53 | 57 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[27,0,0,0,0,46,0,0,0,0,6,6,0,0,67,6],[0,72,0,0,1,0,0,0,0,0,16,53,0,0,53,57] >;
C3×C4.Q8 in GAP, Magma, Sage, TeX
C_3\times C_4.Q_8
% in TeX
G:=Group("C3xC4.Q8"); // GroupNames label
G:=SmallGroup(96,56);
// by ID
G=gap.SmallGroup(96,56);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,1443,117]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^4=b^2,d^2=b^-1*c^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=c^3>;
// generators/relations
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