metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6×D8).8C2, (C3×D4).6D4, (C2×D8).3S3, (C2×C8).34D6, (D4×Dic3)⋊6C2, (C2×D4).61D6, Dic3⋊C8⋊18C2, C6.32(C4○D8), C12.163(C2×D4), C3⋊6(D4.2D4), D4.1(C3⋊D4), C12.92(C4○D4), D4⋊Dic3⋊27C2, C23.12D6⋊3C2, C4.8(D4⋊2S3), C2.27(D8⋊S3), C6.48(C8⋊C22), C2.Dic12⋊16C2, (C2×Dic3).63D4, (C6×D4).80C22, C22.253(S3×D4), C2.16(D8⋊3S3), C6.108(C4⋊D4), (C2×C24).176C22, (C2×C12).430C23, C4⋊Dic3.164C22, (C4×Dic3).44C22, C2.23(C23.14D6), (C2×Dic6).119C22, C4.35(C2×C3⋊D4), (C2×D4.S3)⋊16C2, (C2×C6).343(C2×D4), (C2×C3⋊C8).147C22, (C2×C4).520(C22×S3), SmallGroup(192,712)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×D8).C2
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >
Subgroups: 360 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D4.S3, C6.D4, C2×C24, C3×D8, C2×Dic6, C22×Dic3, C6×D4, D4.2D4, Dic3⋊C8, C2.Dic12, D4⋊Dic3, C2×D4.S3, D4×Dic3, C23.12D6, C6×D8, (C6×D8).C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.2D4, D8⋊S3, D8⋊3S3, C23.14D6, (C6×D8).C2
►On 96 points
Generators in S96
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 95 86)(14 96 87)(15 93 88)(16 94 85)(29 43 36)(30 44 33)(31 41 34)(32 42 35)(37 46 52)(38 47 49)(39 48 50)(40 45 51)(53 61 59)(54 62 60)(55 63 57)(56 64 58)(65 76 69)(66 73 70)(67 74 71)(68 75 72)(77 84 91)(78 81 92)(79 82 89)(80 83 90)
(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 33)(2 36)(3 35)(4 34)(5 31)(6 30)(7 29)(8 32)(9 39)(10 38)(11 37)(12 40)(13 75)(14 74)(15 73)(16 76)(17 42)(18 41)(19 44)(20 43)(21 46)(22 45)(23 48)(24 47)(25 51)(26 50)(27 49)(28 52)(53 92)(54 91)(55 90)(56 89)(57 83)(58 82)(59 81)(60 84)(61 78)(62 77)(63 80)(64 79)(65 85)(66 88)(67 87)(68 86)(69 94)(70 93)(71 96)(72 95)
(1 65 11 59)(2 66 12 60)(3 67 9 57)(4 68 10 58)(5 72 24 64)(6 69 21 61)(7 70 22 62)(8 71 23 63)(13 49 89 41)(14 50 90 42)(15 51 91 43)(16 52 92 44)(17 74 26 55)(18 75 27 56)(19 76 28 53)(20 73 25 54)(29 93 45 77)(30 94 46 78)(31 95 47 79)(32 96 48 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)
(1 57 3 59)(2 60 4 58)(5 64 7 62)(6 63 8 61)(9 65 11 67)(10 68 12 66)(13 52 15 50)(14 51 16 49)(17 53 19 55)(18 56 20 54)(21 71 23 69)(22 70 24 72)(25 73 27 75)(26 76 28 74)(29 78 31 80)(30 77 32 79)(33 84 35 82)(34 83 36 81)(37 88 39 86)(38 87 40 85)(41 90 43 92)(42 89 44 91)(45 94 47 96)(46 93 48 95)
G:=sub<Sym(96)| (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,95,86)(14,96,87)(15,93,88)(16,94,85)(29,43,36)(30,44,33)(31,41,34)(32,42,35)(37,46,52)(38,47,49)(39,48,50)(40,45,51)(53,61,59)(54,62,60)(55,63,57)(56,64,58)(65,76,69)(66,73,70)(67,74,71)(68,75,72)(77,84,91)(78,81,92)(79,82,89)(80,83,90), (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), (1,33)(2,36)(3,35)(4,34)(5,31)(6,30)(7,29)(8,32)(9,39)(10,38)(11,37)(12,40)(13,75)(14,74)(15,73)(16,76)(17,42)(18,41)(19,44)(20,43)(21,46)(22,45)(23,48)(24,47)(25,51)(26,50)(27,49)(28,52)(53,92)(54,91)(55,90)(56,89)(57,83)(58,82)(59,81)(60,84)(61,78)(62,77)(63,80)(64,79)(65,85)(66,88)(67,87)(68,86)(69,94)(70,93)(71,96)(72,95), (1,65,11,59)(2,66,12,60)(3,67,9,57)(4,68,10,58)(5,72,24,64)(6,69,21,61)(7,70,22,62)(8,71,23,63)(13,49,89,41)(14,50,90,42)(15,51,91,43)(16,52,92,44)(17,74,26,55)(18,75,27,56)(19,76,28,53)(20,73,25,54)(29,93,45,77)(30,94,46,78)(31,95,47,79)(32,96,48,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84), (1,57,3,59)(2,60,4,58)(5,64,7,62)(6,63,8,61)(9,65,11,67)(10,68,12,66)(13,52,15,50)(14,51,16,49)(17,53,19,55)(18,56,20,54)(21,71,23,69)(22,70,24,72)(25,73,27,75)(26,76,28,74)(29,78,31,80)(30,77,32,79)(33,84,35,82)(34,83,36,81)(37,88,39,86)(38,87,40,85)(41,90,43,92)(42,89,44,91)(45,94,47,96)(46,93,48,95)>;
G:=Group( (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,95,86)(14,96,87)(15,93,88)(16,94,85)(29,43,36)(30,44,33)(31,41,34)(32,42,35)(37,46,52)(38,47,49)(39,48,50)(40,45,51)(53,61,59)(54,62,60)(55,63,57)(56,64,58)(65,76,69)(66,73,70)(67,74,71)(68,75,72)(77,84,91)(78,81,92)(79,82,89)(80,83,90), (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), (1,33)(2,36)(3,35)(4,34)(5,31)(6,30)(7,29)(8,32)(9,39)(10,38)(11,37)(12,40)(13,75)(14,74)(15,73)(16,76)(17,42)(18,41)(19,44)(20,43)(21,46)(22,45)(23,48)(24,47)(25,51)(26,50)(27,49)(28,52)(53,92)(54,91)(55,90)(56,89)(57,83)(58,82)(59,81)(60,84)(61,78)(62,77)(63,80)(64,79)(65,85)(66,88)(67,87)(68,86)(69,94)(70,93)(71,96)(72,95), (1,65,11,59)(2,66,12,60)(3,67,9,57)(4,68,10,58)(5,72,24,64)(6,69,21,61)(7,70,22,62)(8,71,23,63)(13,49,89,41)(14,50,90,42)(15,51,91,43)(16,52,92,44)(17,74,26,55)(18,75,27,56)(19,76,28,53)(20,73,25,54)(29,93,45,77)(30,94,46,78)(31,95,47,79)(32,96,48,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84), (1,57,3,59)(2,60,4,58)(5,64,7,62)(6,63,8,61)(9,65,11,67)(10,68,12,66)(13,52,15,50)(14,51,16,49)(17,53,19,55)(18,56,20,54)(21,71,23,69)(22,70,24,72)(25,73,27,75)(26,76,28,74)(29,78,31,80)(30,77,32,79)(33,84,35,82)(34,83,36,81)(37,88,39,86)(38,87,40,85)(41,90,43,92)(42,89,44,91)(45,94,47,96)(46,93,48,95) );
G=PermutationGroup([[(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,26),(10,24,27),(11,21,28),(12,22,25),(13,95,86),(14,96,87),(15,93,88),(16,94,85),(29,43,36),(30,44,33),(31,41,34),(32,42,35),(37,46,52),(38,47,49),(39,48,50),(40,45,51),(53,61,59),(54,62,60),(55,63,57),(56,64,58),(65,76,69),(66,73,70),(67,74,71),(68,75,72),(77,84,91),(78,81,92),(79,82,89),(80,83,90)], [(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)], [(1,33),(2,36),(3,35),(4,34),(5,31),(6,30),(7,29),(8,32),(9,39),(10,38),(11,37),(12,40),(13,75),(14,74),(15,73),(16,76),(17,42),(18,41),(19,44),(20,43),(21,46),(22,45),(23,48),(24,47),(25,51),(26,50),(27,49),(28,52),(53,92),(54,91),(55,90),(56,89),(57,83),(58,82),(59,81),(60,84),(61,78),(62,77),(63,80),(64,79),(65,85),(66,88),(67,87),(68,86),(69,94),(70,93),(71,96),(72,95)], [(1,65,11,59),(2,66,12,60),(3,67,9,57),(4,68,10,58),(5,72,24,64),(6,69,21,61),(7,70,22,62),(8,71,23,63),(13,49,89,41),(14,50,90,42),(15,51,91,43),(16,52,92,44),(17,74,26,55),(18,75,27,56),(19,76,28,53),(20,73,25,54),(29,93,45,77),(30,94,46,78),(31,95,47,79),(32,96,48,80),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84)], [(1,57,3,59),(2,60,4,58),(5,64,7,62),(6,63,8,61),(9,65,11,67),(10,68,12,66),(13,52,15,50),(14,51,16,49),(17,53,19,55),(18,56,20,54),(21,71,23,69),(22,70,24,72),(25,73,27,75),(26,76,28,74),(29,78,31,80),(30,77,32,79),(33,84,35,82),(34,83,36,81),(37,88,39,86),(38,87,40,85),(41,90,43,92),(42,89,44,91),(45,94,47,96),(46,93,48,95)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | C8⋊C22 | D4⋊2S3 | S3×D4 | D8⋊S3 | D8⋊3S3 |
| kernel | (C6×D8).C2 | Dic3⋊C8 | C2.Dic12 | D4⋊Dic3 | C2×D4.S3 | D4×Dic3 | C23.12D6 | C6×D8 | C2×D8 | C2×Dic3 | C3×D4 | C2×C8 | C2×D4 | C12 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of (C6×D8).C2 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 71 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 41 | 0 | 0 | 0 | 0 |
| 16 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 48 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 7 |
| 0 | 0 | 0 | 0 | 14 | 7 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 48 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 7 |
| 0 | 0 | 0 | 0 | 14 | 7 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,16,0,0,0,0,41,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,3,0,0,0,0,48,1,0,0,0,0,0,0,66,14,0,0,0,0,7,7],[27,27,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,48,1,0,0,0,0,0,0,66,14,0,0,0,0,7,7] >;
(C6×D8).C2 in GAP, Magma, Sage, TeX
(C_6\times D_8).C_2
% in TeX
G:=Group("(C6xD8).C2"); // GroupNames label
G:=SmallGroup(192,712);
// by ID
G=gap.SmallGroup(192,712);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations