direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×D8.C4, D8.1C12, C12.69D8, C24.101D4, Q16.1C12, (C2×C16)⋊4C6, (C2×C48)⋊8C2, C8.9(C2×C12), C4○D8.1C6, (C3×D8).3C4, C4.18(C3×D8), C8.21(C3×D4), C8.C4⋊1C6, C24.55(C2×C4), (C3×Q16).3C4, (C2×C12).407D4, (C2×C6).17SD16, C6.39(D4⋊C4), C22.1(C3×SD16), C12.71(C22⋊C4), (C2×C24).408C22, (C2×C8).88(C2×C6), (C3×C4○D8).6C2, (C2×C4).61(C3×D4), C4.3(C3×C22⋊C4), C2.8(C3×D4⋊C4), (C3×C8.C4)⋊10C2, SmallGroup(192,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8.C4
G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >
Smallest permutation representation of C3×D8.C4
►On 96 points
Generators in S96
(1 44 28)(2 45 29)(3 46 30)(4 47 31)(5 48 32)(6 41 25)(7 42 26)(8 43 27)(9 84 68)(10 85 69)(11 86 70)(12 87 71)(13 88 72)(14 81 65)(15 82 66)(16 83 67)(17 55 33)(18 56 34)(19 49 35)(20 50 36)(21 51 37)(22 52 38)(23 53 39)(24 54 40)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)(57 61)(58 60)(62 64)(65 67)(68 72)(69 71)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)
(1 71 23 64 5 67 19 60)(2 70 24 63 6 66 20 59)(3 69 17 62 7 65 21 58)(4 68 18 61 8 72 22 57)(9 56 93 43 13 52 89 47)(10 55 94 42 14 51 90 46)(11 54 95 41 15 50 91 45)(12 53 96 48 16 49 92 44)(25 82 36 75 29 86 40 79)(26 81 37 74 30 85 33 78)(27 88 38 73 31 84 34 77)(28 87 39 80 32 83 35 76)
G:=sub<Sym(96)| (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,41,25)(7,42,26)(8,43,27)(9,84,68)(10,85,69)(11,86,70)(12,87,71)(13,88,72)(14,81,65)(15,82,66)(16,83,67)(17,55,33)(18,56,34)(19,49,35)(20,50,36)(21,51,37)(22,52,38)(23,53,39)(24,54,40)(57,89,73)(58,90,74)(59,91,75)(60,92,76)(61,93,77)(62,94,78)(63,95,79)(64,96,80), (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), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,46)(42,45)(43,44)(47,48)(49,56)(50,55)(51,54)(52,53)(57,61)(58,60)(62,64)(65,67)(68,72)(69,71)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96), (1,71,23,64,5,67,19,60)(2,70,24,63,6,66,20,59)(3,69,17,62,7,65,21,58)(4,68,18,61,8,72,22,57)(9,56,93,43,13,52,89,47)(10,55,94,42,14,51,90,46)(11,54,95,41,15,50,91,45)(12,53,96,48,16,49,92,44)(25,82,36,75,29,86,40,79)(26,81,37,74,30,85,33,78)(27,88,38,73,31,84,34,77)(28,87,39,80,32,83,35,76)>;
G:=Group( (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,41,25)(7,42,26)(8,43,27)(9,84,68)(10,85,69)(11,86,70)(12,87,71)(13,88,72)(14,81,65)(15,82,66)(16,83,67)(17,55,33)(18,56,34)(19,49,35)(20,50,36)(21,51,37)(22,52,38)(23,53,39)(24,54,40)(57,89,73)(58,90,74)(59,91,75)(60,92,76)(61,93,77)(62,94,78)(63,95,79)(64,96,80), (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), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,46)(42,45)(43,44)(47,48)(49,56)(50,55)(51,54)(52,53)(57,61)(58,60)(62,64)(65,67)(68,72)(69,71)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96), (1,71,23,64,5,67,19,60)(2,70,24,63,6,66,20,59)(3,69,17,62,7,65,21,58)(4,68,18,61,8,72,22,57)(9,56,93,43,13,52,89,47)(10,55,94,42,14,51,90,46)(11,54,95,41,15,50,91,45)(12,53,96,48,16,49,92,44)(25,82,36,75,29,86,40,79)(26,81,37,74,30,85,33,78)(27,88,38,73,31,84,34,77)(28,87,39,80,32,83,35,76) );
G=PermutationGroup([[(1,44,28),(2,45,29),(3,46,30),(4,47,31),(5,48,32),(6,41,25),(7,42,26),(8,43,27),(9,84,68),(10,85,69),(11,86,70),(12,87,71),(13,88,72),(14,81,65),(15,82,66),(16,83,67),(17,55,33),(18,56,34),(19,49,35),(20,50,36),(21,51,37),(22,52,38),(23,53,39),(24,54,40),(57,89,73),(58,90,74),(59,91,75),(60,92,76),(61,93,77),(62,94,78),(63,95,79),(64,96,80)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,12),(14,16),(17,20),(18,19),(21,24),(22,23),(25,30),(26,29),(27,28),(31,32),(33,36),(34,35),(37,40),(38,39),(41,46),(42,45),(43,44),(47,48),(49,56),(50,55),(51,54),(52,53),(57,61),(58,60),(62,64),(65,67),(68,72),(69,71),(73,77),(74,76),(78,80),(81,83),(84,88),(85,87),(89,93),(90,92),(94,96)], [(1,71,23,64,5,67,19,60),(2,70,24,63,6,66,20,59),(3,69,17,62,7,65,21,58),(4,68,18,61,8,72,22,57),(9,56,93,43,13,52,89,47),(10,55,94,42,14,51,90,46),(11,54,95,41,15,50,91,45),(12,53,96,48,16,49,92,44),(25,82,36,75,29,86,40,79),(26,81,37,74,30,85,33,78),(27,88,38,73,31,84,34,77),(28,87,39,80,32,83,35,76)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 | D8.C4 | C3×D8.C4 |
| kernel | C3×D8.C4 | C3×C8.C4 | C2×C48 | C3×C4○D8 | D8.C4 | C3×D8 | C3×Q16 | C8.C4 | C2×C16 | C4○D8 | D8 | Q16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3×D8.C4 ►in GL2(𝔽97) generated by
| 61 | 0 |
| 0 | 61 |
| 7 | 90 |
| 7 | 7 |
| 7 | 7 |
| 7 | 90 |
| 39 | 94 |
| 94 | 58 |
G:=sub<GL(2,GF(97))| [61,0,0,61],[7,7,90,7],[7,7,7,90],[39,94,94,58] >;
C3×D8.C4 in GAP, Magma, Sage, TeX
C_3\times D_8.C_4
% in TeX
G:=Group("C3xD8.C4"); // GroupNames label
G:=SmallGroup(192,165);
// by ID
G=gap.SmallGroup(192,165);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,850,360,172,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations
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