metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1(C4⋊C4), C6.4(C4⋊D4), C22.61(S3×D4), (C22×C4).32D6, (C22×S3).6Q8, C2.3(Dic3⋊D4), C22.16(S3×Q8), C2.C42⋊3S3, C2.3(D6⋊Q8), (C22×S3).45D4, C6.24(C22⋊Q8), C3⋊1(C23.7Q8), C6.C42⋊36C2, C6.5(C42⋊C2), C2.9(C42⋊2S3), Dic3⋊2(C22⋊C4), (C2×Dic3).130D4, C22.35(C4○D12), (S3×C23).84C22, (C22×C6).292C23, (C22×C12).13C22, C23.267(C22×S3), (C22×Dic3).15C22, (S3×C2×C4)⋊11C4, C6.5(C2×C4⋊C4), C2.7(S3×C4⋊C4), (C2×D6⋊C4).2C2, C6.6(C2×C22⋊C4), C2.8(S3×C22⋊C4), C22.90(S3×C2×C4), (C2×C6).67(C2×Q8), (C2×C4).125(C4×S3), (C2×Dic3⋊C4)⋊1C2, (C2×C6).201(C2×D4), (S3×C22×C4).14C2, (C2×C12).143(C2×C4), (C2×C6).58(C4○D4), (C2×C6).51(C22×C4), (C22×S3).49(C2×C4), (C2×Dic3).78(C2×C4), (C3×C2.C42)⋊20C2, SmallGroup(192,226)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊(C4⋊C4)
G = < a,b,c,d | a6=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a4b, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 640 in 234 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, D6⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.7Q8, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×D6⋊C4, S3×C22×C4, D6⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, S3×C2×C4, C4○D12, S3×D4, S3×Q8, C23.7Q8, C42⋊2S3, S3×C22⋊C4, Dic3⋊D4, S3×C4⋊C4, D6⋊Q8, D6⋊(C4⋊C4)
►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 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 83)(8 82)(9 81)(10 80)(11 79)(12 84)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(31 49)(32 54)(33 53)(34 52)(35 51)(36 50)(37 43)(38 48)(39 47)(40 46)(41 45)(42 44)(55 76)(56 75)(57 74)(58 73)(59 78)(60 77)(61 70)(62 69)(63 68)(64 67)(65 72)(66 71)(85 91)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 65 15 58)(2 64 16 57)(3 63 17 56)(4 62 18 55)(5 61 13 60)(6 66 14 59)(7 48 92 49)(8 47 93 54)(9 46 94 53)(10 45 95 52)(11 44 96 51)(12 43 91 50)(19 70 26 77)(20 69 27 76)(21 68 28 75)(22 67 29 74)(23 72 30 73)(24 71 25 78)(31 79 38 86)(32 84 39 85)(33 83 40 90)(34 82 41 89)(35 81 42 88)(36 80 37 87)
(1 50 26 38)(2 51 27 39)(3 52 28 40)(4 53 29 41)(5 54 30 42)(6 49 25 37)(7 71 87 59)(8 72 88 60)(9 67 89 55)(10 68 90 56)(11 69 85 57)(12 70 86 58)(13 47 23 35)(14 48 24 36)(15 43 19 31)(16 44 20 32)(17 45 21 33)(18 46 22 34)(61 93 73 81)(62 94 74 82)(63 95 75 83)(64 96 76 84)(65 91 77 79)(66 92 78 80)
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), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(31,49)(32,54)(33,53)(34,52)(35,51)(36,50)(37,43)(38,48)(39,47)(40,46)(41,45)(42,44)(55,76)(56,75)(57,74)(58,73)(59,78)(60,77)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92), (1,65,15,58)(2,64,16,57)(3,63,17,56)(4,62,18,55)(5,61,13,60)(6,66,14,59)(7,48,92,49)(8,47,93,54)(9,46,94,53)(10,45,95,52)(11,44,96,51)(12,43,91,50)(19,70,26,77)(20,69,27,76)(21,68,28,75)(22,67,29,74)(23,72,30,73)(24,71,25,78)(31,79,38,86)(32,84,39,85)(33,83,40,90)(34,82,41,89)(35,81,42,88)(36,80,37,87), (1,50,26,38)(2,51,27,39)(3,52,28,40)(4,53,29,41)(5,54,30,42)(6,49,25,37)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,47,23,35)(14,48,24,36)(15,43,19,31)(16,44,20,32)(17,45,21,33)(18,46,22,34)(61,93,73,81)(62,94,74,82)(63,95,75,83)(64,96,76,84)(65,91,77,79)(66,92,78,80)>;
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), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(31,49)(32,54)(33,53)(34,52)(35,51)(36,50)(37,43)(38,48)(39,47)(40,46)(41,45)(42,44)(55,76)(56,75)(57,74)(58,73)(59,78)(60,77)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92), (1,65,15,58)(2,64,16,57)(3,63,17,56)(4,62,18,55)(5,61,13,60)(6,66,14,59)(7,48,92,49)(8,47,93,54)(9,46,94,53)(10,45,95,52)(11,44,96,51)(12,43,91,50)(19,70,26,77)(20,69,27,76)(21,68,28,75)(22,67,29,74)(23,72,30,73)(24,71,25,78)(31,79,38,86)(32,84,39,85)(33,83,40,90)(34,82,41,89)(35,81,42,88)(36,80,37,87), (1,50,26,38)(2,51,27,39)(3,52,28,40)(4,53,29,41)(5,54,30,42)(6,49,25,37)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,47,23,35)(14,48,24,36)(15,43,19,31)(16,44,20,32)(17,45,21,33)(18,46,22,34)(61,93,73,81)(62,94,74,82)(63,95,75,83)(64,96,76,84)(65,91,77,79)(66,92,78,80) );
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)], [(1,21),(2,20),(3,19),(4,24),(5,23),(6,22),(7,83),(8,82),(9,81),(10,80),(11,79),(12,84),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(31,49),(32,54),(33,53),(34,52),(35,51),(36,50),(37,43),(38,48),(39,47),(40,46),(41,45),(42,44),(55,76),(56,75),(57,74),(58,73),(59,78),(60,77),(61,70),(62,69),(63,68),(64,67),(65,72),(66,71),(85,91),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,65,15,58),(2,64,16,57),(3,63,17,56),(4,62,18,55),(5,61,13,60),(6,66,14,59),(7,48,92,49),(8,47,93,54),(9,46,94,53),(10,45,95,52),(11,44,96,51),(12,43,91,50),(19,70,26,77),(20,69,27,76),(21,68,28,75),(22,67,29,74),(23,72,30,73),(24,71,25,78),(31,79,38,86),(32,84,39,85),(33,83,40,90),(34,82,41,89),(35,81,42,88),(36,80,37,87)], [(1,50,26,38),(2,51,27,39),(3,52,28,40),(4,53,29,41),(5,54,30,42),(6,49,25,37),(7,71,87,59),(8,72,88,60),(9,67,89,55),(10,68,90,56),(11,69,85,57),(12,70,86,58),(13,47,23,35),(14,48,24,36),(15,43,19,31),(16,44,20,32),(17,45,21,33),(18,46,22,34),(61,93,73,81),(62,94,74,82),(63,95,75,83),(64,96,76,84),(65,91,77,79),(66,92,78,80)]])
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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | Q8 | D6 | C4○D4 | C4×S3 | C4○D12 | S3×D4 | S3×Q8 |
| kernel | D6⋊(C4⋊C4) | C6.C42 | C3×C2.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | S3×C22×C4 | S3×C2×C4 | C2.C42 | C2×Dic3 | C22×S3 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 1 | 4 | 2 | 2 | 3 | 4 | 4 | 8 | 3 | 1 |
Matrix representation of D6⋊(C4⋊C4) ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,8,0],[8,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,11,9,0,0,0,4,2] >;
D6⋊(C4⋊C4) in GAP, Magma, Sage, TeX
D_6\rtimes (C_4\rtimes C_4)
% in TeX
G:=Group("D6:(C4:C4)"); // GroupNames label
G:=SmallGroup(192,226);
// by ID
G=gap.SmallGroup(192,226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,1094,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations