direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6⋊Q8, C4⋊C4⋊39D6, D6⋊1(C2×Q8), C6⋊2(C22⋊Q8), (C22×S3)⋊4Q8, (C2×C6).52C24, C6.44(C22×D4), C6.24(C22×Q8), C22.33(S3×Q8), D6⋊C4.92C22, Dic3.42(C2×D4), (C22×Dic6)⋊7C2, (C22×C4).194D6, C22.134(S3×D4), (C2×C12).580C23, Dic3⋊C4⋊61C22, (C2×Dic3).190D4, (C2×Dic6)⋊50C22, C22.86(S3×C23), C22.77(C4○D12), (C22×C6).401C23, C23.339(C22×S3), (C22×S3).158C23, (S3×C23).101C22, (C22×C12).434C22, (C2×Dic3).190C23, (C22×Dic3).81C22, C2.7(C2×S3×Q8), (C6×C4⋊C4)⋊14C2, (C2×C4⋊C4)⋊17S3, C2.17(C2×S3×D4), C3⋊2(C2×C22⋊Q8), C6.21(C2×C4○D4), (C2×C6).93(C2×Q8), (C3×C4⋊C4)⋊47C22, (C2×D6⋊C4).18C2, C2.23(C2×C4○D12), (C2×C6).390(C2×D4), (S3×C22×C4).22C2, (C2×Dic3⋊C4)⋊44C2, (S3×C2×C4).288C22, (C2×C6).107(C4○D4), (C2×C4).142(C22×S3), SmallGroup(192,1067)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6⋊Q8
G = < a,b,c,d,e | a2=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d-1 >
Subgroups: 792 in 322 conjugacy classes, 127 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C22⋊Q8, D6⋊Q8, C2×Dic3⋊C4, C2×D6⋊C4, C6×C4⋊C4, C22×Dic6, S3×C22×C4, C2×D6⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C4○D12, S3×D4, S3×Q8, S3×C23, C2×C22⋊Q8, D6⋊Q8, C2×C4○D12, C2×S3×D4, C2×S3×Q8, C2×D6⋊Q8
►On 96 points
Generators in S96
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 92)(8 91)(9 96)(10 95)(11 94)(12 93)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)(43 52)(44 51)(45 50)(46 49)(47 54)(48 53)(55 61)(56 66)(57 65)(58 64)(59 63)(60 62)(67 76)(68 75)(69 74)(70 73)(71 78)(72 77)(79 85)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 28 18 20)(2 27 13 19)(3 26 14 24)(4 25 15 23)(5 30 16 22)(6 29 17 21)(7 89 91 81)(8 88 92 80)(9 87 93 79)(10 86 94 84)(11 85 95 83)(12 90 96 82)(31 53 39 45)(32 52 40 44)(33 51 41 43)(34 50 42 48)(35 49 37 47)(36 54 38 46)(55 72 63 74)(56 71 64 73)(57 70 65 78)(58 69 66 77)(59 68 61 76)(60 67 62 75)
(1 40 18 32)(2 39 13 31)(3 38 14 36)(4 37 15 35)(5 42 16 34)(6 41 17 33)(7 69 91 77)(8 68 92 76)(9 67 93 75)(10 72 94 74)(11 71 95 73)(12 70 96 78)(19 45 27 53)(20 44 28 52)(21 43 29 51)(22 48 30 50)(23 47 25 49)(24 46 26 54)(55 84 63 86)(56 83 64 85)(57 82 65 90)(58 81 66 89)(59 80 61 88)(60 79 62 87)
G:=sub<Sym(96)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,92)(8,91)(9,96)(10,95)(11,94)(12,93)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,61)(56,66)(57,65)(58,64)(59,63)(60,62)(67,76)(68,75)(69,74)(70,73)(71,78)(72,77)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,28,18,20)(2,27,13,19)(3,26,14,24)(4,25,15,23)(5,30,16,22)(6,29,17,21)(7,89,91,81)(8,88,92,80)(9,87,93,79)(10,86,94,84)(11,85,95,83)(12,90,96,82)(31,53,39,45)(32,52,40,44)(33,51,41,43)(34,50,42,48)(35,49,37,47)(36,54,38,46)(55,72,63,74)(56,71,64,73)(57,70,65,78)(58,69,66,77)(59,68,61,76)(60,67,62,75), (1,40,18,32)(2,39,13,31)(3,38,14,36)(4,37,15,35)(5,42,16,34)(6,41,17,33)(7,69,91,77)(8,68,92,76)(9,67,93,75)(10,72,94,74)(11,71,95,73)(12,70,96,78)(19,45,27,53)(20,44,28,52)(21,43,29,51)(22,48,30,50)(23,47,25,49)(24,46,26,54)(55,84,63,86)(56,83,64,85)(57,82,65,90)(58,81,66,89)(59,80,61,88)(60,79,62,87)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,92)(8,91)(9,96)(10,95)(11,94)(12,93)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,61)(56,66)(57,65)(58,64)(59,63)(60,62)(67,76)(68,75)(69,74)(70,73)(71,78)(72,77)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,28,18,20)(2,27,13,19)(3,26,14,24)(4,25,15,23)(5,30,16,22)(6,29,17,21)(7,89,91,81)(8,88,92,80)(9,87,93,79)(10,86,94,84)(11,85,95,83)(12,90,96,82)(31,53,39,45)(32,52,40,44)(33,51,41,43)(34,50,42,48)(35,49,37,47)(36,54,38,46)(55,72,63,74)(56,71,64,73)(57,70,65,78)(58,69,66,77)(59,68,61,76)(60,67,62,75), (1,40,18,32)(2,39,13,31)(3,38,14,36)(4,37,15,35)(5,42,16,34)(6,41,17,33)(7,69,91,77)(8,68,92,76)(9,67,93,75)(10,72,94,74)(11,71,95,73)(12,70,96,78)(19,45,27,53)(20,44,28,52)(21,43,29,51)(22,48,30,50)(23,47,25,49)(24,46,26,54)(55,84,63,86)(56,83,64,85)(57,82,65,90)(58,81,66,89)(59,80,61,88)(60,79,62,87) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,60),(14,55),(15,56),(16,57),(17,58),(18,59),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(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)], [(1,14),(2,13),(3,18),(4,17),(5,16),(6,15),(7,92),(8,91),(9,96),(10,95),(11,94),(12,93),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38),(43,52),(44,51),(45,50),(46,49),(47,54),(48,53),(55,61),(56,66),(57,65),(58,64),(59,63),(60,62),(67,76),(68,75),(69,74),(70,73),(71,78),(72,77),(79,85),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,28,18,20),(2,27,13,19),(3,26,14,24),(4,25,15,23),(5,30,16,22),(6,29,17,21),(7,89,91,81),(8,88,92,80),(9,87,93,79),(10,86,94,84),(11,85,95,83),(12,90,96,82),(31,53,39,45),(32,52,40,44),(33,51,41,43),(34,50,42,48),(35,49,37,47),(36,54,38,46),(55,72,63,74),(56,71,64,73),(57,70,65,78),(58,69,66,77),(59,68,61,76),(60,67,62,75)], [(1,40,18,32),(2,39,13,31),(3,38,14,36),(4,37,15,35),(5,42,16,34),(6,41,17,33),(7,69,91,77),(8,68,92,76),(9,67,93,75),(10,72,94,74),(11,71,95,73),(12,70,96,78),(19,45,27,53),(20,44,28,52),(21,43,29,51),(22,48,30,50),(23,47,25,49),(24,46,26,54),(55,84,63,86),(56,83,64,85),(57,82,65,90),(58,81,66,89),(59,80,61,88),(60,79,62,87)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 | S3×Q8 |
| kernel | C2×D6⋊Q8 | D6⋊Q8 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C4⋊C4 | C22×Dic6 | S3×C22×C4 | C2×C4⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C22×C4 | C2×C6 | C22 | C22 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 4 | 8 | 2 | 2 |
Matrix representation of C2×D6⋊Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 10 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,10,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×D6⋊Q8 in GAP, Magma, Sage, TeX
C_2\times D_6\rtimes Q_8
% in TeX
G:=Group("C2xD6:Q8"); // GroupNames label
G:=SmallGroup(192,1067);
// by ID
G=gap.SmallGroup(192,1067);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^-1>;
// generators/relations