metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊6C4, D6⋊4(C4⋊C4), C6.68(C4×D4), C6.41C22≀C2, (C2×C12).251D4, (C22×C4).58D6, C2.4(C23⋊2D6), C22.27(S3×Q8), C2.7(D6⋊Q8), C2.4(D6⋊3Q8), (C22×S3).89D4, C22.112(S3×D4), C6.49(C22⋊Q8), (C22×S3).10Q8, C3⋊4(C23.8Q8), C6.C42⋊40C2, (C2×Dic3).177D4, C2.6(D6.D4), C2.19(Dic3⋊5D4), C22.58(C4○D12), (S3×C23).90C22, C23.307(C22×S3), (C22×C6).353C23, (C22×C12).347C22, C22.29(Q8⋊3S3), C6.51(C22.D4), (C22×Dic3).59C22, (C2×C4⋊C4)⋊7S3, (C2×C4)⋊4(C4×S3), (C6×C4⋊C4)⋊24C2, C6.21(C2×C4⋊C4), C2.22(S3×C4⋊C4), (C2×C12)⋊21(C2×C4), C2.13(C4×C3⋊D4), (C2×C6).84(C2×Q8), (C2×D6⋊C4).12C2, (C2×Dic3)⋊7(C2×C4), (C2×C6).334(C2×D4), (S3×C22×C4).19C2, C22.138(S3×C2×C4), (C2×Dic3⋊C4)⋊13C2, C22.68(C2×C3⋊D4), (C2×C6).190(C4○D4), (C2×C4).169(C3⋊D4), (C22×S3).42(C2×C4), (C2×C6).121(C22×C4), SmallGroup(192,548)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C4⋊6C4
G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 632 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×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C23.8Q8, C6.C42, C2×Dic3⋊C4, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, D6⋊C4⋊6C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, S3×C2×C4, C4○D12, S3×D4, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.8Q8, S3×C4⋊C4, Dic3⋊5D4, D6.D4, D6⋊Q8, C4×C3⋊D4, C23⋊2D6, D6⋊3Q8, D6⋊C4⋊6C4
►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 81)(8 80)(9 79)(10 84)(11 83)(12 82)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)(31 50)(32 49)(33 54)(34 53)(35 52)(36 51)(37 44)(38 43)(39 48)(40 47)(41 46)(42 45)(55 77)(56 76)(57 75)(58 74)(59 73)(60 78)(61 71)(62 70)(63 69)(64 68)(65 67)(66 72)(85 95)(86 94)(87 93)(88 92)(89 91)(90 96)
(1 64 16 60)(2 65 17 55)(3 66 18 56)(4 61 13 57)(5 62 14 58)(6 63 15 59)(7 51 95 47)(8 52 96 48)(9 53 91 43)(10 54 92 44)(11 49 93 45)(12 50 94 46)(19 77 29 67)(20 78 30 68)(21 73 25 69)(22 74 26 70)(23 75 27 71)(24 76 28 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)
(1 48 24 36)(2 43 19 31)(3 44 20 32)(4 45 21 33)(5 46 22 34)(6 47 23 35)(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 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(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), (1,27)(2,26)(3,25)(4,30)(5,29)(6,28)(7,81)(8,80)(9,79)(10,84)(11,83)(12,82)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)(31,50)(32,49)(33,54)(34,53)(35,52)(36,51)(37,44)(38,43)(39,48)(40,47)(41,46)(42,45)(55,77)(56,76)(57,75)(58,74)(59,73)(60,78)(61,71)(62,70)(63,69)(64,68)(65,67)(66,72)(85,95)(86,94)(87,93)(88,92)(89,91)(90,96), (1,64,16,60)(2,65,17,55)(3,66,18,56)(4,61,13,57)(5,62,14,58)(6,63,15,59)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84), (1,48,24,36)(2,43,19,31)(3,44,20,32)(4,45,21,33)(5,46,22,34)(6,47,23,35)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(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), (1,27)(2,26)(3,25)(4,30)(5,29)(6,28)(7,81)(8,80)(9,79)(10,84)(11,83)(12,82)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)(31,50)(32,49)(33,54)(34,53)(35,52)(36,51)(37,44)(38,43)(39,48)(40,47)(41,46)(42,45)(55,77)(56,76)(57,75)(58,74)(59,73)(60,78)(61,71)(62,70)(63,69)(64,68)(65,67)(66,72)(85,95)(86,94)(87,93)(88,92)(89,91)(90,96), (1,64,16,60)(2,65,17,55)(3,66,18,56)(4,61,13,57)(5,62,14,58)(6,63,15,59)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84), (1,48,24,36)(2,43,19,31)(3,44,20,32)(4,45,21,33)(5,46,22,34)(6,47,23,35)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(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)], [(1,27),(2,26),(3,25),(4,30),(5,29),(6,28),(7,81),(8,80),(9,79),(10,84),(11,83),(12,82),(13,20),(14,19),(15,24),(16,23),(17,22),(18,21),(31,50),(32,49),(33,54),(34,53),(35,52),(36,51),(37,44),(38,43),(39,48),(40,47),(41,46),(42,45),(55,77),(56,76),(57,75),(58,74),(59,73),(60,78),(61,71),(62,70),(63,69),(64,68),(65,67),(66,72),(85,95),(86,94),(87,93),(88,92),(89,91),(90,96)], [(1,64,16,60),(2,65,17,55),(3,66,18,56),(4,61,13,57),(5,62,14,58),(6,63,15,59),(7,51,95,47),(8,52,96,48),(9,53,91,43),(10,54,92,44),(11,49,93,45),(12,50,94,46),(19,77,29,67),(20,78,30,68),(21,73,25,69),(22,74,26,70),(23,75,27,71),(24,76,28,72),(31,89,41,79),(32,90,42,80),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84)], [(1,48,24,36),(2,43,19,31),(3,44,20,32),(4,45,21,33),(5,46,22,34),(6,47,23,35),(7,71,87,59),(8,72,88,60),(9,67,89,55),(10,68,90,56),(11,69,85,57),(12,70,86,58),(13,49,25,37),(14,50,26,38),(15,51,27,39),(16,52,28,40),(17,53,29,41),(18,54,30,42),(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 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | Q8 | D6 | C4○D4 | C4×S3 | C3⋊D4 | C4○D12 | S3×D4 | S3×Q8 | Q8⋊3S3 |
| kernel | D6⋊C4⋊6C4 | C6.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C4⋊C4 | S3×C22×C4 | D6⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 2 | 1 | 1 |
Matrix representation of D6⋊C4⋊6C4 ►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 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 11 | 9 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 | 5 |
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],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,1],[12,0,0,0,0,0,11,4,0,0,0,9,2,0,0,0,0,0,12,1,0,0,0,11,1],[5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,8,5,0,0,0,0,5] >;
D6⋊C4⋊6C4 in GAP, Magma, Sage, TeX
D_6\rtimes C_4\rtimes_6C_4
% in TeX
G:=Group("D6:C4:6C4"); // GroupNames label
G:=SmallGroup(192,548);
// by ID
G=gap.SmallGroup(192,548);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,387,58,6278]);
// 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=c^-1>;
// generators/relations