direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.C8, D4.C24, Q8.2C24, C24.107D4, M5(2)⋊4C6, M4(2).3C12, (C2×C48)⋊6C2, (C2×C16)⋊2C6, C4.3(C2×C24), C8○D4.4C6, (C3×D4).3C8, C8.27(C3×D4), (C3×Q8).3C8, C12.32(C2×C8), C4○D4.4C12, (C2×C6).8M4(2), C6.27(C22⋊C8), (C3×M5(2))⋊12C2, (C3×M4(2)).7C4, (C2×C24).442C22, C12.113(C22⋊C4), C22.1(C3×M4(2)), (C2×C8).96(C2×C6), (C3×C4○D4).5C4, (C3×C8○D4).5C2, C2.8(C3×C22⋊C8), (C2×C4).42(C2×C12), C4.30(C3×C22⋊C4), (C2×C12).263(C2×C4), SmallGroup(192,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.C8
G = < a,b,c,d | a3=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >
Smallest permutation representation of C3×D4.C8
►On 96 points
Generators in S96
(1 53 82)(2 54 83)(3 55 84)(4 56 85)(5 57 86)(6 58 87)(7 59 88)(8 60 89)(9 61 90)(10 62 91)(11 63 92)(12 64 93)(13 49 94)(14 50 95)(15 51 96)(16 52 81)(17 72 47)(18 73 48)(19 74 33)(20 75 34)(21 76 35)(22 77 36)(23 78 37)(24 79 38)(25 80 39)(26 65 40)(27 66 41)(28 67 42)(29 68 43)(30 69 44)(31 70 45)(32 71 46)
(1 69 9 77)(2 70 10 78)(3 71 11 79)(4 72 12 80)(5 73 13 65)(6 74 14 66)(7 75 15 67)(8 76 16 68)(17 93 25 85)(18 94 26 86)(19 95 27 87)(20 96 28 88)(21 81 29 89)(22 82 30 90)(23 83 31 91)(24 84 32 92)(33 50 41 58)(34 51 42 59)(35 52 43 60)(36 53 44 61)(37 54 45 62)(38 55 46 63)(39 56 47 64)(40 57 48 49)
(1 77)(2 10)(3 71)(5 65)(6 14)(7 75)(9 69)(11 79)(13 73)(15 67)(17 25)(18 94)(20 88)(21 29)(22 82)(24 92)(26 86)(28 96)(30 90)(32 84)(34 59)(35 43)(36 53)(38 63)(39 47)(40 57)(42 51)(44 61)(46 55)(48 49)(50 58)(54 62)(68 76)(72 80)(83 91)(87 95)
(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)
G:=sub<Sym(96)| (1,53,82)(2,54,83)(3,55,84)(4,56,85)(5,57,86)(6,58,87)(7,59,88)(8,60,89)(9,61,90)(10,62,91)(11,63,92)(12,64,93)(13,49,94)(14,50,95)(15,51,96)(16,52,81)(17,72,47)(18,73,48)(19,74,33)(20,75,34)(21,76,35)(22,77,36)(23,78,37)(24,79,38)(25,80,39)(26,65,40)(27,66,41)(28,67,42)(29,68,43)(30,69,44)(31,70,45)(32,71,46), (1,69,9,77)(2,70,10,78)(3,71,11,79)(4,72,12,80)(5,73,13,65)(6,74,14,66)(7,75,15,67)(8,76,16,68)(17,93,25,85)(18,94,26,86)(19,95,27,87)(20,96,28,88)(21,81,29,89)(22,82,30,90)(23,83,31,91)(24,84,32,92)(33,50,41,58)(34,51,42,59)(35,52,43,60)(36,53,44,61)(37,54,45,62)(38,55,46,63)(39,56,47,64)(40,57,48,49), (1,77)(2,10)(3,71)(5,65)(6,14)(7,75)(9,69)(11,79)(13,73)(15,67)(17,25)(18,94)(20,88)(21,29)(22,82)(24,92)(26,86)(28,96)(30,90)(32,84)(34,59)(35,43)(36,53)(38,63)(39,47)(40,57)(42,51)(44,61)(46,55)(48,49)(50,58)(54,62)(68,76)(72,80)(83,91)(87,95), (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)>;
G:=Group( (1,53,82)(2,54,83)(3,55,84)(4,56,85)(5,57,86)(6,58,87)(7,59,88)(8,60,89)(9,61,90)(10,62,91)(11,63,92)(12,64,93)(13,49,94)(14,50,95)(15,51,96)(16,52,81)(17,72,47)(18,73,48)(19,74,33)(20,75,34)(21,76,35)(22,77,36)(23,78,37)(24,79,38)(25,80,39)(26,65,40)(27,66,41)(28,67,42)(29,68,43)(30,69,44)(31,70,45)(32,71,46), (1,69,9,77)(2,70,10,78)(3,71,11,79)(4,72,12,80)(5,73,13,65)(6,74,14,66)(7,75,15,67)(8,76,16,68)(17,93,25,85)(18,94,26,86)(19,95,27,87)(20,96,28,88)(21,81,29,89)(22,82,30,90)(23,83,31,91)(24,84,32,92)(33,50,41,58)(34,51,42,59)(35,52,43,60)(36,53,44,61)(37,54,45,62)(38,55,46,63)(39,56,47,64)(40,57,48,49), (1,77)(2,10)(3,71)(5,65)(6,14)(7,75)(9,69)(11,79)(13,73)(15,67)(17,25)(18,94)(20,88)(21,29)(22,82)(24,92)(26,86)(28,96)(30,90)(32,84)(34,59)(35,43)(36,53)(38,63)(39,47)(40,57)(42,51)(44,61)(46,55)(48,49)(50,58)(54,62)(68,76)(72,80)(83,91)(87,95), (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) );
G=PermutationGroup([[(1,53,82),(2,54,83),(3,55,84),(4,56,85),(5,57,86),(6,58,87),(7,59,88),(8,60,89),(9,61,90),(10,62,91),(11,63,92),(12,64,93),(13,49,94),(14,50,95),(15,51,96),(16,52,81),(17,72,47),(18,73,48),(19,74,33),(20,75,34),(21,76,35),(22,77,36),(23,78,37),(24,79,38),(25,80,39),(26,65,40),(27,66,41),(28,67,42),(29,68,43),(30,69,44),(31,70,45),(32,71,46)], [(1,69,9,77),(2,70,10,78),(3,71,11,79),(4,72,12,80),(5,73,13,65),(6,74,14,66),(7,75,15,67),(8,76,16,68),(17,93,25,85),(18,94,26,86),(19,95,27,87),(20,96,28,88),(21,81,29,89),(22,82,30,90),(23,83,31,91),(24,84,32,92),(33,50,41,58),(34,51,42,59),(35,52,43,60),(36,53,44,61),(37,54,45,62),(38,55,46,63),(39,56,47,64),(40,57,48,49)], [(1,77),(2,10),(3,71),(5,65),(6,14),(7,75),(9,69),(11,79),(13,73),(15,67),(17,25),(18,94),(20,88),(21,29),(22,82),(24,92),(26,86),(28,96),(30,90),(32,84),(34,59),(35,43),(36,53),(38,63),(39,47),(40,57),(42,51),(44,61),(46,55),(48,49),(50,58),(54,62),(68,76),(72,80),(83,91),(87,95)], [(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)]])
84 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 | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 24M | 24N | 24O | 24P | 48A | ··· | 48P | 48Q | ··· | 48X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | D4 | M4(2) | C3×D4 | C3×M4(2) | D4.C8 | C3×D4.C8 |
| kernel | C3×D4.C8 | C2×C48 | C3×M5(2) | C3×C8○D4 | D4.C8 | C3×M4(2) | C3×C4○D4 | C2×C16 | M5(2) | C8○D4 | C3×D4 | C3×Q8 | M4(2) | C4○D4 | D4 | Q8 | C24 | C2×C6 | C8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3×D4.C8 ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 33 |
| 0 | 47 | 0 |
| 96 | 0 | 0 |
| 0 | 0 | 33 |
| 0 | 50 | 0 |
| 22 | 0 | 0 |
| 0 | 5 | 68 |
| 0 | 41 | 5 |
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,0,47,0,33,0],[96,0,0,0,0,50,0,33,0],[22,0,0,0,5,41,0,68,5] >;
C3×D4.C8 in GAP, Magma, Sage, TeX
C_3\times D_4.C_8
% in TeX
G:=Group("C3xD4.C8"); // GroupNames label
G:=SmallGroup(192,156);
// by ID
G=gap.SmallGroup(192,156);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1522,248,2111,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations
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