metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊3C8, C42.201D6, C3⋊3(C8×D4), C4⋊1(S3×C8), C3⋊C8⋊30D4, C4⋊C8⋊18S3, C12⋊2(C2×C8), D6⋊3(C2×C8), D6⋊C8⋊27C2, C6.46(C4×D4), D6⋊C4.11C4, (C4×D12).7C2, C4.205(S3×D4), (C2×C8).216D6, (C2×D12).12C4, C6.28(C8○D4), C12.364(C2×D4), C4⋊Dic3.18C4, C6.11(C22×C8), C2.4(D12.C4), (C4×C12).60C22, C12.334(C4○D4), C2.2(Dic3⋊5D4), (C2×C12).831C23, (C2×C24).253C22, C4.53(Q8⋊3S3), (C4×C3⋊C8)⋊4C2, (S3×C2×C8)⋊22C2, C2.13(S3×C2×C8), (C3×C4⋊C8)⋊20C2, (C2×C4).72(C4×S3), C22.48(S3×C2×C4), (C2×C12).158(C2×C4), (C2×C3⋊C8).306C22, (S3×C2×C4).277C22, (C2×C6).86(C22×C4), (C22×S3).37(C2×C4), (C2×C4).773(C22×S3), (C2×Dic3).53(C2×C4), SmallGroup(192,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊C8
G = < a,b,c | a12=b2=c8=1, bab=a-1, cac-1=a7, cbc-1=a6b >
Subgroups: 312 in 134 conjugacy classes, 61 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, S3×C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C8×D4, C4×C3⋊C8, D6⋊C8, C3×C4⋊C8, C4×D12, S3×C2×C8, D12⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C2×C8, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C22×C8, C8○D4, S3×C8, S3×C2×C4, S3×D4, Q8⋊3S3, C8×D4, Dic3⋊5D4, S3×C2×C8, D12.C4, D12⋊C8
►On 96 points
(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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 48)(49 51)(52 60)(53 59)(54 58)(55 57)(61 71)(62 70)(63 69)(64 68)(65 67)(73 83)(74 82)(75 81)(76 80)(77 79)(85 91)(86 90)(87 89)(92 96)(93 95)
(1 68 22 35 77 43 93 58)(2 63 23 30 78 38 94 53)(3 70 24 25 79 45 95 60)(4 65 13 32 80 40 96 55)(5 72 14 27 81 47 85 50)(6 67 15 34 82 42 86 57)(7 62 16 29 83 37 87 52)(8 69 17 36 84 44 88 59)(9 64 18 31 73 39 89 54)(10 71 19 26 74 46 90 49)(11 66 20 33 75 41 91 56)(12 61 21 28 76 48 92 51)
G:=sub<Sym(96)| (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,83)(74,82)(75,81)(76,80)(77,79)(85,91)(86,90)(87,89)(92,96)(93,95), (1,68,22,35,77,43,93,58)(2,63,23,30,78,38,94,53)(3,70,24,25,79,45,95,60)(4,65,13,32,80,40,96,55)(5,72,14,27,81,47,85,50)(6,67,15,34,82,42,86,57)(7,62,16,29,83,37,87,52)(8,69,17,36,84,44,88,59)(9,64,18,31,73,39,89,54)(10,71,19,26,74,46,90,49)(11,66,20,33,75,41,91,56)(12,61,21,28,76,48,92,51)>;
G:=Group( (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,83)(74,82)(75,81)(76,80)(77,79)(85,91)(86,90)(87,89)(92,96)(93,95), (1,68,22,35,77,43,93,58)(2,63,23,30,78,38,94,53)(3,70,24,25,79,45,95,60)(4,65,13,32,80,40,96,55)(5,72,14,27,81,47,85,50)(6,67,15,34,82,42,86,57)(7,62,16,29,83,37,87,52)(8,69,17,36,84,44,88,59)(9,64,18,31,73,39,89,54)(10,71,19,26,74,46,90,49)(11,66,20,33,75,41,91,56)(12,61,21,28,76,48,92,51) );
G=PermutationGroup([[(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24),(25,29),(26,28),(30,36),(31,35),(32,34),(37,45),(38,44),(39,43),(40,42),(46,48),(49,51),(52,60),(53,59),(54,58),(55,57),(61,71),(62,70),(63,69),(64,68),(65,67),(73,83),(74,82),(75,81),(76,80),(77,79),(85,91),(86,90),(87,89),(92,96),(93,95)], [(1,68,22,35,77,43,93,58),(2,63,23,30,78,38,94,53),(3,70,24,25,79,45,95,60),(4,65,13,32,80,40,96,55),(5,72,14,27,81,47,85,50),(6,67,15,34,82,42,86,57),(7,62,16,29,83,37,87,52),(8,69,17,36,84,44,88,59),(9,64,18,31,73,39,89,54),(10,71,19,26,74,46,90,49),(11,66,20,33,75,41,91,56),(12,61,21,28,76,48,92,51)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D6 | C4○D4 | C4×S3 | C8○D4 | S3×C8 | S3×D4 | Q8⋊3S3 | D12.C4 |
| kernel | D12⋊C8 | C4×C3⋊C8 | D6⋊C8 | C3×C4⋊C8 | C4×D12 | S3×C2×C8 | C4⋊Dic3 | D6⋊C4 | C2×D12 | D12 | C4⋊C8 | C3⋊C8 | C42 | C2×C8 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of D12⋊C8 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 29 | 2 |
| 0 | 0 | 17 | 44 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 29 | 1 |
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 44 | 71 |
| 0 | 0 | 55 | 29 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,29,17,0,0,2,44],[72,0,0,0,72,1,0,0,0,0,72,29,0,0,0,1],[22,0,0,0,0,22,0,0,0,0,44,55,0,0,71,29] >;
D12⋊C8 in GAP, Magma, Sage, TeX
D_{12}\rtimes C_8 % in TeX
G:=Group("D12:C8"); // GroupNames label
G:=SmallGroup(192,393);
// by ID
G=gap.SmallGroup(192,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^6*b>;
// generators/relations