direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C2.D8, C24⋊3C4, C8⋊1C12, C6.14D8, C6.7Q16, C12.10Q8, C4⋊C4.3C6, (C2×C8).3C6, C2.2(C3×D8), C4.2(C3×Q8), C4.7(C2×C12), (C2×C24).9C2, (C2×C6).49D4, C6.13(C4⋊C4), C2.2(C3×Q16), C12.44(C2×C4), C22.11(C3×D4), (C2×C12).118C22, C2.4(C3×C4⋊C4), (C3×C4⋊C4).10C2, (C2×C4).21(C2×C6), SmallGroup(96,57)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2.D8
G = < a,b,c,d | a3=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C3×C2.D8
►Regular action on 96 points
Generators in S96
(1 11 35)(2 12 36)(3 13 37)(4 14 38)(5 15 39)(6 16 40)(7 9 33)(8 10 34)(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 92 72)(26 93 65)(27 94 66)(28 95 67)(29 96 68)(30 89 69)(31 90 70)(32 91 71)(57 78 85)(58 79 86)(59 80 87)(60 73 88)(61 74 81)(62 75 82)(63 76 83)(64 77 84)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(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 72 19 58)(2 71 20 57)(3 70 21 64)(4 69 22 63)(5 68 23 62)(6 67 24 61)(7 66 17 60)(8 65 18 59)(9 27 53 73)(10 26 54 80)(11 25 55 79)(12 32 56 78)(13 31 49 77)(14 30 50 76)(15 29 51 75)(16 28 52 74)(33 94 41 88)(34 93 42 87)(35 92 43 86)(36 91 44 85)(37 90 45 84)(38 89 46 83)(39 96 47 82)(40 95 48 81)
G:=sub<Sym(96)| (1,11,35)(2,12,36)(3,13,37)(4,14,38)(5,15,39)(6,16,40)(7,9,33)(8,10,34)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,92,72)(26,93,65)(27,94,66)(28,95,67)(29,96,68)(30,89,69)(31,90,70)(32,91,71)(57,78,85)(58,79,86)(59,80,87)(60,73,88)(61,74,81)(62,75,82)(63,76,83)(64,77,84), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(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), (1,72,19,58)(2,71,20,57)(3,70,21,64)(4,69,22,63)(5,68,23,62)(6,67,24,61)(7,66,17,60)(8,65,18,59)(9,27,53,73)(10,26,54,80)(11,25,55,79)(12,32,56,78)(13,31,49,77)(14,30,50,76)(15,29,51,75)(16,28,52,74)(33,94,41,88)(34,93,42,87)(35,92,43,86)(36,91,44,85)(37,90,45,84)(38,89,46,83)(39,96,47,82)(40,95,48,81)>;
G:=Group( (1,11,35)(2,12,36)(3,13,37)(4,14,38)(5,15,39)(6,16,40)(7,9,33)(8,10,34)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,92,72)(26,93,65)(27,94,66)(28,95,67)(29,96,68)(30,89,69)(31,90,70)(32,91,71)(57,78,85)(58,79,86)(59,80,87)(60,73,88)(61,74,81)(62,75,82)(63,76,83)(64,77,84), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(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), (1,72,19,58)(2,71,20,57)(3,70,21,64)(4,69,22,63)(5,68,23,62)(6,67,24,61)(7,66,17,60)(8,65,18,59)(9,27,53,73)(10,26,54,80)(11,25,55,79)(12,32,56,78)(13,31,49,77)(14,30,50,76)(15,29,51,75)(16,28,52,74)(33,94,41,88)(34,93,42,87)(35,92,43,86)(36,91,44,85)(37,90,45,84)(38,89,46,83)(39,96,47,82)(40,95,48,81) );
G=PermutationGroup([[(1,11,35),(2,12,36),(3,13,37),(4,14,38),(5,15,39),(6,16,40),(7,9,33),(8,10,34),(17,53,41),(18,54,42),(19,55,43),(20,56,44),(21,49,45),(22,50,46),(23,51,47),(24,52,48),(25,92,72),(26,93,65),(27,94,66),(28,95,67),(29,96,68),(30,89,69),(31,90,70),(32,91,71),(57,78,85),(58,79,86),(59,80,87),(60,73,88),(61,74,81),(62,75,82),(63,76,83),(64,77,84)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(25,79),(26,80),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(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)], [(1,72,19,58),(2,71,20,57),(3,70,21,64),(4,69,22,63),(5,68,23,62),(6,67,24,61),(7,66,17,60),(8,65,18,59),(9,27,53,73),(10,26,54,80),(11,25,55,79),(12,32,56,78),(13,31,49,77),(14,30,50,76),(15,29,51,75),(16,28,52,74),(33,94,41,88),(34,93,42,87),(35,92,43,86),(36,91,44,85),(37,90,45,84),(38,89,46,83),(39,96,47,82),(40,95,48,81)]])
C3×C2.D8 is a maximal subgroup of
C6.6D16 C6.SD32 C6.D16 C6.Q32 Dic3⋊5D8 Dic3⋊5Q16 C24⋊2Q8 Dic3.Q16 C24⋊4Q8 Dic6.2Q8 C8.6Dic6 C8.27(C4×S3) C8⋊S3⋊C4 D6.5D8 D6⋊2D8 D6.2Q16 C2.D8⋊S3 C8⋊3D12 D6⋊2Q16 C2.D8⋊7S3 C24⋊C2⋊C4 D12⋊2Q8 D12.2Q8 C12×D8 C12×Q16
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 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | D8 | Q16 | C3×Q8 | C3×D4 | C3×D8 | C3×Q16 |
| kernel | C3×C2.D8 | C3×C4⋊C4 | C2×C24 | C2.D8 | C24 | C4⋊C4 | C2×C8 | C8 | C12 | C2×C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C2.D8 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 49 | 70 | 0 | 0 | 0 | 0 |
| 70 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 56 | 0 | 0 |
| 0 | 0 | 56 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 62 |
| 0 | 0 | 0 | 0 | 62 | 30 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[49,70,0,0,0,0,70,24,0,0,0,0,0,0,32,56,0,0,0,0,56,41,0,0,0,0,0,0,43,62,0,0,0,0,62,30] >;
C3×C2.D8 in GAP, Magma, Sage, TeX
C_3\times C_2.D_8
% in TeX
G:=Group("C3xC2.D8"); // GroupNames label
G:=SmallGroup(96,57);
// by ID
G=gap.SmallGroup(96,57);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,367,1443,117]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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