metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊2C4, C6.3(C4×D4), D6⋊2(C4⋊C4), C2.6(C4×D12), C6.3C22≀C2, (C2×C4).112D12, (C2×C12).234D4, (C22×C4).33D6, C22.62(S3×D4), (C22×S3).7Q8, C2.2(D6⋊D4), C22.17(S3×Q8), C6.C42⋊2C2, C2.C42⋊9S3, C2.4(D6⋊Q8), C2.2(C4.D12), (C22×S3).67D4, C22.25(C2×D12), C6.25(C22⋊Q8), C3⋊1(C23.8Q8), (C2×Dic3).131D4, C2.4(C23.9D6), C2.8(Dic3⋊4D4), C22.36(C4○D12), (S3×C23).85C22, C23.268(C22×S3), (C22×C6).293C23, C6.9(C22.D4), C22.37(D4⋊2S3), (C22×C12).333C22, (C22×Dic3).16C22, (C2×C4)⋊3(C4×S3), C6.6(C2×C4⋊C4), C2.8(S3×C4⋊C4), (C2×C12)⋊5(C2×C4), (C2×D6⋊C4).3C2, (C2×C4⋊Dic3)⋊1C2, C22.91(S3×C2×C4), (C2×C6).68(C2×Q8), (C2×Dic3)⋊3(C2×C4), (C2×C6).202(C2×D4), (S3×C22×C4).15C2, (C2×Dic3⋊C4)⋊31C2, (C2×C6).52(C22×C4), (C2×C6).132(C4○D4), (C22×S3).30(C2×C4), (C3×C2.C42)⋊15C2, SmallGroup(192,227)
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=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >
Subgroups: 640 in 234 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, D6, C2×C6, C22⋊C4, C4⋊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, C4⋊Dic3, D6⋊C4, D6⋊C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C23.8Q8, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, S3×C22×C4, D6⋊C4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C22×S3, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, C23.8Q8, C4×D12, Dic3⋊4D4, D6⋊D4, C23.9D6, S3×C4⋊C4, D6⋊Q8, C4.D12, 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 27)(2 26)(3 25)(4 30)(5 29)(6 28)(7 83)(8 82)(9 81)(10 80)(11 79)(12 84)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(31 52)(32 51)(33 50)(34 49)(35 54)(36 53)(37 46)(38 45)(39 44)(40 43)(41 48)(42 47)(55 73)(56 78)(57 77)(58 76)(59 75)(60 74)(61 67)(62 72)(63 71)(64 70)(65 69)(66 68)(85 91)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 61 13 56)(2 62 14 57)(3 63 15 58)(4 64 16 59)(5 65 17 60)(6 66 18 55)(7 51 94 46)(8 52 95 47)(9 53 96 48)(10 54 91 43)(11 49 92 44)(12 50 93 45)(19 78 30 67)(20 73 25 68)(21 74 26 69)(22 75 27 70)(23 76 28 71)(24 77 29 72)(31 90 42 79)(32 85 37 80)(33 86 38 81)(34 87 39 82)(35 88 40 83)(36 89 41 84)
(1 44 20 32)(2 45 21 33)(3 46 22 34)(4 47 23 35)(5 48 24 36)(6 43 19 31)(7 67 87 55)(8 68 88 56)(9 69 89 57)(10 70 90 58)(11 71 85 59)(12 72 86 60)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 95 73 83)(62 96 74 84)(63 91 75 79)(64 92 76 80)(65 93 77 81)(66 94 78 82)
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), (1,27)(2,26)(3,25)(4,30)(5,29)(6,28)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)(31,52)(32,51)(33,50)(34,49)(35,54)(36,53)(37,46)(38,45)(39,44)(40,43)(41,48)(42,47)(55,73)(56,78)(57,77)(58,76)(59,75)(60,74)(61,67)(62,72)(63,71)(64,70)(65,69)(66,68)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92), (1,61,13,56)(2,62,14,57)(3,63,15,58)(4,64,16,59)(5,65,17,60)(6,66,18,55)(7,51,94,46)(8,52,95,47)(9,53,96,48)(10,54,91,43)(11,49,92,44)(12,50,93,45)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82)>;
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), (1,27)(2,26)(3,25)(4,30)(5,29)(6,28)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)(31,52)(32,51)(33,50)(34,49)(35,54)(36,53)(37,46)(38,45)(39,44)(40,43)(41,48)(42,47)(55,73)(56,78)(57,77)(58,76)(59,75)(60,74)(61,67)(62,72)(63,71)(64,70)(65,69)(66,68)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92), (1,61,13,56)(2,62,14,57)(3,63,15,58)(4,64,16,59)(5,65,17,60)(6,66,18,55)(7,51,94,46)(8,52,95,47)(9,53,96,48)(10,54,91,43)(11,49,92,44)(12,50,93,45)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82) );
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)], [(1,27),(2,26),(3,25),(4,30),(5,29),(6,28),(7,83),(8,82),(9,81),(10,80),(11,79),(12,84),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23),(31,52),(32,51),(33,50),(34,49),(35,54),(36,53),(37,46),(38,45),(39,44),(40,43),(41,48),(42,47),(55,73),(56,78),(57,77),(58,76),(59,75),(60,74),(61,67),(62,72),(63,71),(64,70),(65,69),(66,68),(85,91),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,61,13,56),(2,62,14,57),(3,63,15,58),(4,64,16,59),(5,65,17,60),(6,66,18,55),(7,51,94,46),(8,52,95,47),(9,53,96,48),(10,54,91,43),(11,49,92,44),(12,50,93,45),(19,78,30,67),(20,73,25,68),(21,74,26,69),(22,75,27,70),(23,76,28,71),(24,77,29,72),(31,90,42,79),(32,85,37,80),(33,86,38,81),(34,87,39,82),(35,88,40,83),(36,89,41,84)], [(1,44,20,32),(2,45,21,33),(3,46,22,34),(4,47,23,35),(5,48,24,36),(6,43,19,31),(7,67,87,55),(8,68,88,56),(9,69,89,57),(10,70,90,58),(11,71,85,59),(12,72,86,60),(13,49,25,37),(14,50,26,38),(15,51,27,39),(16,52,28,40),(17,53,29,41),(18,54,30,42),(61,95,73,83),(62,96,74,84),(63,91,75,79),(64,92,76,80),(65,93,77,81),(66,94,78,82)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | Q8 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | D6⋊C4⋊C4 | C6.C42 | C3×C2.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×D6⋊C4 | S3×C22×C4 | D6⋊C4 | C2.C42 | C2×Dic3 | C2×C12 | C22×S3 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 2 | 1 | 1 |
Matrix representation of D6⋊C4⋊C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 10 | 6 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 9 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,1,4,0,0,0,0,12],[1,0,0,0,0,0,10,6,0,0,0,7,3,0,0,0,0,0,7,10,0,0,0,3,6],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,9,0,0,0,0,1] >;
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(192,227);
// by ID
G=gap.SmallGroup(192,227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations