metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊4D8, C8⋊4D12, C4⋊1D24, C24⋊22D4, C42.262D6, (C4×C8)⋊8S3, C6.4(C2×D8), (C4×C24)⋊13C2, (C2×D24)⋊1C2, C3⋊1(C8⋊4D4), C2.6(C2×D24), C4⋊D12⋊1C2, C4.31(C2×D12), (C2×C4).81D12, (C2×C8).301D6, (C2×C12).378D4, C12.274(C2×D4), C6.4(C4⋊1D4), C2.6(C4⋊D12), (C2×D12).3C22, C22.91(C2×D12), (C2×C12).724C23, (C2×C24).374C22, (C4×C12).308C22, (C2×C6).107(C2×D4), (C2×C4).667(C22×S3), SmallGroup(192,254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊4D8
G = < a,b,c | a12=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 728 in 162 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, D6, C2×C6, C42, C2×C8, D8, C2×D4, C24, D12, C2×C12, C2×C12, C22×S3, C4×C8, C4⋊1D4, C2×D8, D24, C4×C12, C2×C24, C2×D12, C2×D12, C8⋊4D4, C4×C24, C4⋊D12, C2×D24, C12⋊4D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C4⋊1D4, C2×D8, D24, C2×D12, C8⋊4D4, C4⋊D12, C2×D24, C12⋊4D8
►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 63 76 15 46 95 36 51)(2 64 77 16 47 96 25 52)(3 65 78 17 48 85 26 53)(4 66 79 18 37 86 27 54)(5 67 80 19 38 87 28 55)(6 68 81 20 39 88 29 56)(7 69 82 21 40 89 30 57)(8 70 83 22 41 90 31 58)(9 71 84 23 42 91 32 59)(10 72 73 24 43 92 33 60)(11 61 74 13 44 93 34 49)(12 62 75 14 45 94 35 50)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 29)(14 28)(15 27)(16 26)(17 25)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 84)(59 83)(60 82)
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), (1,63,76,15,46,95,36,51)(2,64,77,16,47,96,25,52)(3,65,78,17,48,85,26,53)(4,66,79,18,37,86,27,54)(5,67,80,19,38,87,28,55)(6,68,81,20,39,88,29,56)(7,69,82,21,40,89,30,57)(8,70,83,22,41,90,31,58)(9,71,84,23,42,91,32,59)(10,72,73,24,43,92,33,60)(11,61,74,13,44,93,34,49)(12,62,75,14,45,94,35,50), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,29)(14,28)(15,27)(16,26)(17,25)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82)>;
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), (1,63,76,15,46,95,36,51)(2,64,77,16,47,96,25,52)(3,65,78,17,48,85,26,53)(4,66,79,18,37,86,27,54)(5,67,80,19,38,87,28,55)(6,68,81,20,39,88,29,56)(7,69,82,21,40,89,30,57)(8,70,83,22,41,90,31,58)(9,71,84,23,42,91,32,59)(10,72,73,24,43,92,33,60)(11,61,74,13,44,93,34,49)(12,62,75,14,45,94,35,50), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,29)(14,28)(15,27)(16,26)(17,25)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82) );
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)], [(1,63,76,15,46,95,36,51),(2,64,77,16,47,96,25,52),(3,65,78,17,48,85,26,53),(4,66,79,18,37,86,27,54),(5,67,80,19,38,87,28,55),(6,68,81,20,39,88,29,56),(7,69,82,21,40,89,30,57),(8,70,83,22,41,90,31,58),(9,71,84,23,42,91,32,59),(10,72,73,24,43,92,33,60),(11,61,74,13,44,93,34,49),(12,62,75,14,45,94,35,50)], [(1,66),(2,65),(3,64),(4,63),(5,62),(6,61),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,29),(14,28),(15,27),(16,26),(17,25),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,96),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,84),(59,83),(60,82)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 24 | 24 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D12 | D12 | D24 |
| kernel | C12⋊4D8 | C4×C24 | C4⋊D12 | C2×D24 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C12⋊4D8 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 23 | 5 | 0 | 0 |
| 68 | 18 | 0 | 0 |
| 0 | 0 | 55 | 5 |
| 0 | 0 | 68 | 50 |
| 50 | 68 | 0 | 0 |
| 18 | 23 | 0 | 0 |
| 0 | 0 | 50 | 68 |
| 0 | 0 | 18 | 23 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,0,72,0,0,1,72],[23,68,0,0,5,18,0,0,0,0,55,68,0,0,5,50],[50,18,0,0,68,23,0,0,0,0,50,18,0,0,68,23] >;
C12⋊4D8 in GAP, Magma, Sage, TeX
C_{12}\rtimes_4D_8 % in TeX
G:=Group("C12:4D8"); // GroupNames label
G:=SmallGroup(192,254);
// by ID
G=gap.SmallGroup(192,254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations