direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊Dic3, C23.35D6, C22.15D12, C22.5Dic6, C6⋊2(C4⋊C4), C12⋊7(C2×C4), (C2×C12)⋊5C4, C6.9(C2×Q8), (C2×C6).6Q8, (C2×C4)⋊3Dic3, C4⋊2(C2×Dic3), C6.15(C2×D4), C2.2(C2×D12), (C2×C6).20D4, (C2×C4).84D6, (C22×C4).8S3, C2.3(C2×Dic6), (C2×C6).43C23, C6.23(C22×C4), (C22×C12).7C2, (C2×C12).92C22, C2.4(C22×Dic3), (C22×C6).35C22, C22.21(C22×S3), (C22×Dic3).5C2, C22.14(C2×Dic3), (C2×Dic3).34C22, C3⋊3(C2×C4⋊C4), (C2×C6).34(C2×C4), SmallGroup(96,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊Dic3
G = < a,b,c,d | a2=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 162 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C2×C4⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C2×C4⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C2×C4⋊Dic3
►Regular action on 96 points
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 71)(8 72)(9 67)(10 68)(11 69)(12 70)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(19 54)(20 49)(21 50)(22 51)(23 52)(24 53)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)(55 90)(56 85)(57 86)(58 87)(59 88)(60 89)(61 80)(62 81)(63 82)(64 83)(65 84)(66 79)(73 92)(74 93)(75 94)(76 95)(77 96)(78 91)
(1 29 17 22)(2 30 18 23)(3 25 13 24)(4 26 14 19)(5 27 15 20)(6 28 16 21)(7 80 92 88)(8 81 93 89)(9 82 94 90)(10 83 95 85)(11 84 96 86)(12 79 91 87)(31 51 38 43)(32 52 39 44)(33 53 40 45)(34 54 41 46)(35 49 42 47)(36 50 37 48)(55 67 63 75)(56 68 64 76)(57 69 65 77)(58 70 66 78)(59 71 61 73)(60 72 62 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 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)
G:=sub<Sym(96)| (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,71)(8,72)(9,67)(10,68)(11,69)(12,70)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(19,54)(20,49)(21,50)(22,51)(23,52)(24,53)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(55,90)(56,85)(57,86)(58,87)(59,88)(60,89)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(73,92)(74,93)(75,94)(76,95)(77,96)(78,91), (1,29,17,22)(2,30,18,23)(3,25,13,24)(4,26,14,19)(5,27,15,20)(6,28,16,21)(7,80,92,88)(8,81,93,89)(9,82,94,90)(10,83,95,85)(11,84,96,86)(12,79,91,87)(31,51,38,43)(32,52,39,44)(33,53,40,45)(34,54,41,46)(35,49,42,47)(36,50,37,48)(55,67,63,75)(56,68,64,76)(57,69,65,77)(58,70,66,78)(59,71,61,73)(60,72,62,74), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,65,16,62)(14,64,17,61)(15,63,18,66)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,89,40,86)(38,88,41,85)(39,87,42,90)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
G:=Group( (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,71)(8,72)(9,67)(10,68)(11,69)(12,70)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(19,54)(20,49)(21,50)(22,51)(23,52)(24,53)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(55,90)(56,85)(57,86)(58,87)(59,88)(60,89)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(73,92)(74,93)(75,94)(76,95)(77,96)(78,91), (1,29,17,22)(2,30,18,23)(3,25,13,24)(4,26,14,19)(5,27,15,20)(6,28,16,21)(7,80,92,88)(8,81,93,89)(9,82,94,90)(10,83,95,85)(11,84,96,86)(12,79,91,87)(31,51,38,43)(32,52,39,44)(33,53,40,45)(34,54,41,46)(35,49,42,47)(36,50,37,48)(55,67,63,75)(56,68,64,76)(57,69,65,77)(58,70,66,78)(59,71,61,73)(60,72,62,74), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,65,16,62)(14,64,17,61)(15,63,18,66)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,89,40,86)(38,88,41,85)(39,87,42,90)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
G=PermutationGroup([[(1,38),(2,39),(3,40),(4,41),(5,42),(6,37),(7,71),(8,72),(9,67),(10,68),(11,69),(12,70),(13,33),(14,34),(15,35),(16,36),(17,31),(18,32),(19,54),(20,49),(21,50),(22,51),(23,52),(24,53),(25,45),(26,46),(27,47),(28,48),(29,43),(30,44),(55,90),(56,85),(57,86),(58,87),(59,88),(60,89),(61,80),(62,81),(63,82),(64,83),(65,84),(66,79),(73,92),(74,93),(75,94),(76,95),(77,96),(78,91)], [(1,29,17,22),(2,30,18,23),(3,25,13,24),(4,26,14,19),(5,27,15,20),(6,28,16,21),(7,80,92,88),(8,81,93,89),(9,82,94,90),(10,83,95,85),(11,84,96,86),(12,79,91,87),(31,51,38,43),(32,52,39,44),(33,53,40,45),(34,54,41,46),(35,49,42,47),(36,50,37,48),(55,67,63,75),(56,68,64,76),(57,69,65,77),(58,70,66,78),(59,71,61,73),(60,72,62,74)], [(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)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,65,16,62),(14,64,17,61),(15,63,18,66),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,80,34,83),(32,79,35,82),(33,84,36,81),(37,89,40,86),(38,88,41,85),(39,87,42,90),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
C2×C4⋊Dic3 is a maximal subgroup of
C12.C42 C12.9C42 M4(2)⋊Dic3 C6.(C4×Q8) C2.(C4×D12) C2.(C4×Dic6) Dic3⋊C4⋊C4 (C2×C4)⋊Dic6 C6.(C4⋊Q8) (C2×C4).17D12 (C2×C4).Dic6 (C22×C4).85D6 D6⋊C4⋊C4 D6⋊C4⋊3C4 (C2×C4).21D12 (C2×C12).33D4 C23.39D12 C23.40D12 C23.43D12 C22.D24 C12⋊4(C4⋊C4) (C2×Dic6)⋊7C4 C42⋊10Dic3 C42⋊11Dic3 (C2×C4)⋊6D12 C24.17D6 C24.18D6 C24.58D6 C24.19D6 C24.21D6 C24.27D6 C12⋊(C4⋊C4) Dic3×C4⋊C4 (C4×Dic3)⋊8C4 (C4×Dic3)⋊9C4 C4⋊C4⋊5Dic3 (C2×C4).44D12 (C2×C12).54D4 C4⋊C4⋊6Dic3 (C2×C12).55D4 C4⋊(D6⋊C4) (C2×C12).56D4 C4⋊C4.232D6 (C2×C6).D8 C4⋊D4.S3 (C2×Q8).49D6 (C2×C6).Q16 C23.52D12 C23.54D12 C24.75D6 C24.30D6 (C6×Q8)⋊7C4 C4○D4⋊3Dic3 C2×C4×Dic6 C2×C4×D12 C2×S3×C4⋊C4 C42.90D6 C42.91D6 C42.105D6 D4⋊6Dic6 D4⋊6D12 C42.119D6 C6.732- 1+4 C6.1152+ 1+4 C6.1182+ 1+4 C6.772- 1+4 C6.852- 1+4 C2×D4×Dic3 C2×Q8×Dic3 C6.1442+ 1+4 C6.1082- 1+4
C2×C4⋊Dic3 is a maximal quotient of
C12⋊7M4(2) C42⋊10Dic3 C42⋊11Dic3 C24.58D6 C4⋊C4⋊6Dic3 C42.43D6 C23.27D12 C23.52D12 C23.9Dic6 C24.75D6
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D6 | Dic6 | D12 |
| kernel | C2×C4⋊Dic3 | C4⋊Dic3 | C22×Dic3 | C22×C12 | C2×C12 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 4 |
Matrix representation of C2×C4⋊Dic3 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 7 |
| 0 | 0 | 6 | 10 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 1 |
| 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 2 | 2 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,3,6,0,0,7,10],[12,0,0,0,0,1,0,0,0,0,0,12,0,0,1,1],[5,0,0,0,0,1,0,0,0,0,11,2,0,0,4,2] >;
C2×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xC4:Dic3"); // GroupNames label
G:=SmallGroup(96,132);
// by ID
G=gap.SmallGroup(96,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations