direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×C3⋊C8, C42.205D6, C3⋊4(C8×D4), C12⋊3(C2×C8), (C3×D4)⋊2C8, (C6×D4).8C4, C6.90(C4×D4), C12⋊C8⋊20C2, (C4×D4).15S3, (D4×C12).5C2, C4.214(S3×D4), C2.3(D4×Dic3), C6.38(C8○D4), C12.373(C2×D4), C4⋊C4.11Dic3, C6.24(C22×C8), (C4×C12).81C22, (C2×D4).11Dic3, (C22×C4).325D6, C22⋊C4.7Dic3, C12.306(C4○D4), C12.55D4⋊24C2, (C2×C12).848C23, C2.2(D4.Dic3), C4.133(D4⋊2S3), C23.21(C2×Dic3), (C22×C12).348C22, C22.22(C22×Dic3), C4⋊1(C2×C3⋊C8), (C4×C3⋊C8)⋊6C2, (C2×C6)⋊3(C2×C8), C22⋊2(C2×C3⋊C8), C2.5(C22×C3⋊C8), (C3×C4⋊C4).10C4, (C22×C3⋊C8)⋊18C2, (C3×C22⋊C4).8C4, (C2×C12).162(C2×C4), (C2×C3⋊C8).315C22, (C22×C6).59(C2×C4), (C2×C4).33(C2×Dic3), (C2×C6).185(C22×C4), (C2×C4).790(C22×S3), SmallGroup(192,569)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C3⋊C8
G = < a,b,c,d | a4=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 232 in 134 conjugacy classes, 77 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C8×D4, C4×C3⋊C8, C12⋊C8, C12.55D4, C22×C3⋊C8, D4×C12, D4×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, Dic3, D6, C2×C8, C22×C4, C2×D4, C4○D4, C3⋊C8, C2×Dic3, C22×S3, C4×D4, C22×C8, C8○D4, C2×C3⋊C8, S3×D4, D4⋊2S3, C22×Dic3, C8×D4, C22×C3⋊C8, D4×Dic3, D4.Dic3, D4×C3⋊C8
►On 96 points
(1 55 22 87)(2 56 23 88)(3 49 24 81)(4 50 17 82)(5 51 18 83)(6 52 19 84)(7 53 20 85)(8 54 21 86)(9 65 77 33)(10 66 78 34)(11 67 79 35)(12 68 80 36)(13 69 73 37)(14 70 74 38)(15 71 75 39)(16 72 76 40)(25 41 93 57)(26 42 94 58)(27 43 95 59)(28 44 96 60)(29 45 89 61)(30 46 90 62)(31 47 91 63)(32 48 92 64)
(1 5)(2 6)(3 7)(4 8)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)(49 85)(50 86)(51 87)(52 88)(53 81)(54 82)(55 83)(56 84)(65 69)(66 70)(67 71)(68 72)(89 93)(90 94)(91 95)(92 96)
(1 69 27)(2 28 70)(3 71 29)(4 30 72)(5 65 31)(6 32 66)(7 67 25)(8 26 68)(9 63 83)(10 84 64)(11 57 85)(12 86 58)(13 59 87)(14 88 60)(15 61 81)(16 82 62)(17 90 40)(18 33 91)(19 92 34)(20 35 93)(21 94 36)(22 37 95)(23 96 38)(24 39 89)(41 53 79)(42 80 54)(43 55 73)(44 74 56)(45 49 75)(46 76 50)(47 51 77)(48 78 52)
(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)| (1,55,22,87)(2,56,23,88)(3,49,24,81)(4,50,17,82)(5,51,18,83)(6,52,19,84)(7,53,20,85)(8,54,21,86)(9,65,77,33)(10,66,78,34)(11,67,79,35)(12,68,80,36)(13,69,73,37)(14,70,74,38)(15,71,75,39)(16,72,76,40)(25,41,93,57)(26,42,94,58)(27,43,95,59)(28,44,96,60)(29,45,89,61)(30,46,90,62)(31,47,91,63)(32,48,92,64), (1,5)(2,6)(3,7)(4,8)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60)(49,85)(50,86)(51,87)(52,88)(53,81)(54,82)(55,83)(56,84)(65,69)(66,70)(67,71)(68,72)(89,93)(90,94)(91,95)(92,96), (1,69,27)(2,28,70)(3,71,29)(4,30,72)(5,65,31)(6,32,66)(7,67,25)(8,26,68)(9,63,83)(10,84,64)(11,57,85)(12,86,58)(13,59,87)(14,88,60)(15,61,81)(16,82,62)(17,90,40)(18,33,91)(19,92,34)(20,35,93)(21,94,36)(22,37,95)(23,96,38)(24,39,89)(41,53,79)(42,80,54)(43,55,73)(44,74,56)(45,49,75)(46,76,50)(47,51,77)(48,78,52), (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( (1,55,22,87)(2,56,23,88)(3,49,24,81)(4,50,17,82)(5,51,18,83)(6,52,19,84)(7,53,20,85)(8,54,21,86)(9,65,77,33)(10,66,78,34)(11,67,79,35)(12,68,80,36)(13,69,73,37)(14,70,74,38)(15,71,75,39)(16,72,76,40)(25,41,93,57)(26,42,94,58)(27,43,95,59)(28,44,96,60)(29,45,89,61)(30,46,90,62)(31,47,91,63)(32,48,92,64), (1,5)(2,6)(3,7)(4,8)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60)(49,85)(50,86)(51,87)(52,88)(53,81)(54,82)(55,83)(56,84)(65,69)(66,70)(67,71)(68,72)(89,93)(90,94)(91,95)(92,96), (1,69,27)(2,28,70)(3,71,29)(4,30,72)(5,65,31)(6,32,66)(7,67,25)(8,26,68)(9,63,83)(10,84,64)(11,57,85)(12,86,58)(13,59,87)(14,88,60)(15,61,81)(16,82,62)(17,90,40)(18,33,91)(19,92,34)(20,35,93)(21,94,36)(22,37,95)(23,96,38)(24,39,89)(41,53,79)(42,80,54)(43,55,73)(44,74,56)(45,49,75)(46,76,50)(47,51,77)(48,78,52), (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([[(1,55,22,87),(2,56,23,88),(3,49,24,81),(4,50,17,82),(5,51,18,83),(6,52,19,84),(7,53,20,85),(8,54,21,86),(9,65,77,33),(10,66,78,34),(11,67,79,35),(12,68,80,36),(13,69,73,37),(14,70,74,38),(15,71,75,39),(16,72,76,40),(25,41,93,57),(26,42,94,58),(27,43,95,59),(28,44,96,60),(29,45,89,61),(30,46,90,62),(31,47,91,63),(32,48,92,64)], [(1,5),(2,6),(3,7),(4,8),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60),(49,85),(50,86),(51,87),(52,88),(53,81),(54,82),(55,83),(56,84),(65,69),(66,70),(67,71),(68,72),(89,93),(90,94),(91,95),(92,96)], [(1,69,27),(2,28,70),(3,71,29),(4,30,72),(5,65,31),(6,32,66),(7,67,25),(8,26,68),(9,63,83),(10,84,64),(11,57,85),(12,86,58),(13,59,87),(14,88,60),(15,61,81),(16,82,62),(17,90,40),(18,33,91),(19,92,34),(20,35,93),(21,94,36),(22,37,95),(23,96,38),(24,39,89),(41,53,79),(42,80,54),(43,55,73),(44,74,56),(45,49,75),(46,76,50),(47,51,77),(48,78,52)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 8I | ··· | 8T | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D4 | D6 | Dic3 | Dic3 | D6 | Dic3 | C4○D4 | C3⋊C8 | C8○D4 | S3×D4 | D4⋊2S3 | D4.Dic3 |
| kernel | D4×C3⋊C8 | C4×C3⋊C8 | C12⋊C8 | C12.55D4 | C22×C3⋊C8 | D4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C3×D4 | C4×D4 | C3⋊C8 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 16 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 4 | 1 | 1 | 2 |
Matrix representation of D4×C3⋊C8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 44 | 71 | 0 | 0 |
| 42 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[44,42,0,0,71,29,0,0,0,0,1,0,0,0,0,1] >;
D4×C3⋊C8 in GAP, Magma, Sage, TeX
D_4\times C_3\rtimes C_8
% in TeX
G:=Group("D4xC3:C8"); // GroupNames label
G:=SmallGroup(192,569);
// by ID
G=gap.SmallGroup(192,569);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,219,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations