direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6.D8, C4⋊C4⋊35D6, (C2×D12)⋊9C4, (C2×C6).39D8, C6.48(C2×D8), D12⋊19(C2×C4), C4.7(D6⋊C4), C4.59(C2×D12), C6⋊1(D4⋊C4), C12.139(C2×D4), (C2×C12).133D4, (C2×C4).139D12, (C2×C6).40SD16, C6.66(C2×SD16), C12.59(C22×C4), (C22×C6).183D4, (C22×C4).345D6, C12.19(C22⋊C4), (C2×C12).318C23, C22.45(D6⋊C4), C22.20(D4⋊S3), (C22×D12).10C2, (C2×D12).232C22, C23.104(C3⋊D4), (C22×C12).133C22, C22.11(Q8⋊2S3), (C6×C4⋊C4)⋊1C2, (C2×C4⋊C4)⋊1S3, C4.48(S3×C2×C4), C3⋊2(C2×D4⋊C4), (C22×C3⋊C8)⋊1C2, C2.2(C2×D4⋊S3), (C2×C3⋊C8)⋊31C22, (C2×C4).75(C4×S3), C2.11(C2×D6⋊C4), (C3×C4⋊C4)⋊40C22, (C2×C12).77(C2×C4), (C2×C6).438(C2×D4), C6.38(C2×C22⋊C4), C2.2(C2×Q8⋊2S3), C22.57(C2×C3⋊D4), (C2×C4).124(C3⋊D4), (C2×C6).57(C22⋊C4), (C2×C4).418(C22×S3), SmallGroup(192,524)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6.D8
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >
Subgroups: 696 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C12, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, C3⋊C8, D12, D12, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C2×C3⋊C8, C2×C3⋊C8, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×D12, C22×C12, C22×C12, S3×C23, C2×D4⋊C4, C6.D8, C22×C3⋊C8, C6×C4⋊C4, C22×D12, C2×C6.D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, D6⋊C4, D4⋊S3, Q8⋊2S3, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D4⋊C4, C6.D8, C2×D6⋊C4, C2×D4⋊S3, C2×Q8⋊2S3, C2×C6.D8
►On 96 points
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 65)(48 66)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 84)
(1 83 10 47 62 53)(2 54 63 48 11 84)(3 85 12 41 64 55)(4 56 57 42 13 86)(5 87 14 43 58 49)(6 50 59 44 15 88)(7 81 16 45 60 51)(8 52 61 46 9 82)(17 75 94 69 32 34)(18 35 25 70 95 76)(19 77 96 71 26 36)(20 37 27 72 89 78)(21 79 90 65 28 38)(22 39 29 66 91 80)(23 73 92 67 30 40)(24 33 31 68 93 74)
(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 17)(2 68)(3 23)(4 66)(5 21)(6 72)(7 19)(8 70)(9 76)(10 32)(11 74)(12 30)(13 80)(14 28)(15 78)(16 26)(18 46)(20 44)(22 42)(24 48)(25 52)(27 50)(29 56)(31 54)(33 63)(34 83)(35 61)(36 81)(37 59)(38 87)(39 57)(40 85)(41 67)(43 65)(45 71)(47 69)(49 79)(51 77)(53 75)(55 73)(58 90)(60 96)(62 94)(64 92)(82 95)(84 93)(86 91)(88 89)
G:=sub<Sym(96)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84), (1,83,10,47,62,53)(2,54,63,48,11,84)(3,85,12,41,64,55)(4,56,57,42,13,86)(5,87,14,43,58,49)(6,50,59,44,15,88)(7,81,16,45,60,51)(8,52,61,46,9,82)(17,75,94,69,32,34)(18,35,25,70,95,76)(19,77,96,71,26,36)(20,37,27,72,89,78)(21,79,90,65,28,38)(22,39,29,66,91,80)(23,73,92,67,30,40)(24,33,31,68,93,74), (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), (1,17)(2,68)(3,23)(4,66)(5,21)(6,72)(7,19)(8,70)(9,76)(10,32)(11,74)(12,30)(13,80)(14,28)(15,78)(16,26)(18,46)(20,44)(22,42)(24,48)(25,52)(27,50)(29,56)(31,54)(33,63)(34,83)(35,61)(36,81)(37,59)(38,87)(39,57)(40,85)(41,67)(43,65)(45,71)(47,69)(49,79)(51,77)(53,75)(55,73)(58,90)(60,96)(62,94)(64,92)(82,95)(84,93)(86,91)(88,89)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84), (1,83,10,47,62,53)(2,54,63,48,11,84)(3,85,12,41,64,55)(4,56,57,42,13,86)(5,87,14,43,58,49)(6,50,59,44,15,88)(7,81,16,45,60,51)(8,52,61,46,9,82)(17,75,94,69,32,34)(18,35,25,70,95,76)(19,77,96,71,26,36)(20,37,27,72,89,78)(21,79,90,65,28,38)(22,39,29,66,91,80)(23,73,92,67,30,40)(24,33,31,68,93,74), (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), (1,17)(2,68)(3,23)(4,66)(5,21)(6,72)(7,19)(8,70)(9,76)(10,32)(11,74)(12,30)(13,80)(14,28)(15,78)(16,26)(18,46)(20,44)(22,42)(24,48)(25,52)(27,50)(29,56)(31,54)(33,63)(34,83)(35,61)(36,81)(37,59)(38,87)(39,57)(40,85)(41,67)(43,65)(45,71)(47,69)(49,79)(51,77)(53,75)(55,73)(58,90)(60,96)(62,94)(64,92)(82,95)(84,93)(86,91)(88,89) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,65),(48,66),(73,85),(74,86),(75,87),(76,88),(77,81),(78,82),(79,83),(80,84)], [(1,83,10,47,62,53),(2,54,63,48,11,84),(3,85,12,41,64,55),(4,56,57,42,13,86),(5,87,14,43,58,49),(6,50,59,44,15,88),(7,81,16,45,60,51),(8,52,61,46,9,82),(17,75,94,69,32,34),(18,35,25,70,95,76),(19,77,96,71,26,36),(20,37,27,72,89,78),(21,79,90,65,28,38),(22,39,29,66,91,80),(23,73,92,67,30,40),(24,33,31,68,93,74)], [(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)], [(1,17),(2,68),(3,23),(4,66),(5,21),(6,72),(7,19),(8,70),(9,76),(10,32),(11,74),(12,30),(13,80),(14,28),(15,78),(16,26),(18,46),(20,44),(22,42),(24,48),(25,52),(27,50),(29,56),(31,54),(33,63),(34,83),(35,61),(36,81),(37,59),(38,87),(39,57),(40,85),(41,67),(43,65),(45,71),(47,69),(49,79),(51,77),(53,75),(55,73),(58,90),(60,96),(62,94),(64,92),(82,95),(84,93),(86,91),(88,89)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | D4⋊S3 | Q8⋊2S3 |
| kernel | C2×C6.D8 | C6.D8 | C22×C3⋊C8 | C6×C4⋊C4 | C22×D12 | C2×D12 | C2×C4⋊C4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C2×C6.D8 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,16,57,0,0,0,0,16,16],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×C6.D8 in GAP, Magma, Sage, TeX
C_2\times C_6.D_8
% in TeX
G:=Group("C2xC6.D8"); // GroupNames label
G:=SmallGroup(192,524);
// by ID
G=gap.SmallGroup(192,524);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
// generators/relations