metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊3C4, C6.25(C4×D4), C2.7(C4×D12), (C2×C4).113D12, (C2×C12).235D4, C22.64(S3×D4), (C22×C4).34D6, C2.2(C12⋊D4), C6.34(C4⋊D4), C6.C42⋊4C2, (C2×Dic3).85D4, C22.26(C2×D12), C2.C42⋊10S3, C6.19(C4.4D4), (S3×C23).2C22, C6.30(C42⋊C2), C6.21(C42⋊2C2), C22.38(C4○D12), (C22×C6).295C23, C23.270(C22×S3), (C22×C12).15C22, C3⋊2(C24.C22), C2.10(Dic3⋊4D4), C22.39(D4⋊2S3), C2.4(C23.11D6), C22.20(Q8⋊3S3), C2.2(C23.21D6), C6.10(C22.D4), (C22×Dic3).177C22, (C2×C4).27(C4×S3), (C2×D6⋊C4).5C2, (C2×C4⋊Dic3)⋊2C2, (C2×C4×Dic3)⋊18C2, C22.93(S3×C2×C4), (C2×C12).35(C2×C4), (C2×C6).204(C2×D4), C2.4(C4⋊C4⋊S3), C2.8(C4⋊C4⋊7S3), (C22×S3).8(C2×C4), (C2×C6).54(C22×C4), (C2×C6).134(C4○D4), (C2×Dic3).44(C2×C4), (C3×C2.C42)⋊16C2, SmallGroup(192,229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C4⋊3C4
G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd-1=a3bc2, dcd-1=a3c-1 >
Subgroups: 544 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C24.C22, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×C4⋊Dic3, C2×D6⋊C4, D6⋊C4⋊3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C24.C22, C4×D12, Dic3⋊4D4, C23.11D6, C23.21D6, C4⋊C4⋊7S3, C12⋊D4, C4⋊C4⋊S3, D6⋊C4⋊3C4
►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 6)(2 5)(3 4)(7 93)(8 92)(9 91)(10 96)(11 95)(12 94)(13 18)(14 17)(15 16)(19 22)(20 21)(23 24)(25 30)(26 29)(27 28)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(56 60)(57 59)(61 63)(64 66)(68 72)(69 71)(73 75)(76 78)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 61 13 60)(2 62 14 55)(3 63 15 56)(4 64 16 57)(5 65 17 58)(6 66 18 59)(7 51 92 44)(8 52 93 45)(9 53 94 46)(10 54 95 47)(11 49 96 48)(12 50 91 43)(19 74 26 67)(20 75 27 68)(21 76 28 69)(22 77 29 70)(23 78 30 71)(24 73 25 72)(31 86 38 79)(32 87 39 80)(33 88 40 81)(34 89 41 82)(35 90 42 83)(36 85 37 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), (1,6)(2,5)(3,4)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(13,18)(14,17)(15,16)(19,22)(20,21)(23,24)(25,30)(26,29)(27,28)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(56,60)(57,59)(61,63)(64,66)(68,72)(69,71)(73,75)(76,78)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,61,13,60)(2,62,14,55)(3,63,15,56)(4,64,16,57)(5,65,17,58)(6,66,18,59)(7,51,92,44)(8,52,93,45)(9,53,94,46)(10,54,95,47)(11,49,96,48)(12,50,91,43)(19,74,26,67)(20,75,27,68)(21,76,28,69)(22,77,29,70)(23,78,30,71)(24,73,25,72)(31,86,38,79)(32,87,39,80)(33,88,40,81)(34,89,41,82)(35,90,42,83)(36,85,37,84), (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), (1,6)(2,5)(3,4)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(13,18)(14,17)(15,16)(19,22)(20,21)(23,24)(25,30)(26,29)(27,28)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(56,60)(57,59)(61,63)(64,66)(68,72)(69,71)(73,75)(76,78)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,61,13,60)(2,62,14,55)(3,63,15,56)(4,64,16,57)(5,65,17,58)(6,66,18,59)(7,51,92,44)(8,52,93,45)(9,53,94,46)(10,54,95,47)(11,49,96,48)(12,50,91,43)(19,74,26,67)(20,75,27,68)(21,76,28,69)(22,77,29,70)(23,78,30,71)(24,73,25,72)(31,86,38,79)(32,87,39,80)(33,88,40,81)(34,89,41,82)(35,90,42,83)(36,85,37,84), (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)], [(1,6),(2,5),(3,4),(7,93),(8,92),(9,91),(10,96),(11,95),(12,94),(13,18),(14,17),(15,16),(19,22),(20,21),(23,24),(25,30),(26,29),(27,28),(31,38),(32,37),(33,42),(34,41),(35,40),(36,39),(43,50),(44,49),(45,54),(46,53),(47,52),(48,51),(56,60),(57,59),(61,63),(64,66),(68,72),(69,71),(73,75),(76,78),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90)], [(1,61,13,60),(2,62,14,55),(3,63,15,56),(4,64,16,57),(5,65,17,58),(6,66,18,59),(7,51,92,44),(8,52,93,45),(9,53,94,46),(10,54,95,47),(11,49,96,48),(12,50,91,43),(19,74,26,67),(20,75,27,68),(21,76,28,69),(22,77,29,70),(23,78,30,71),(24,73,25,72),(31,86,38,79),(32,87,39,80),(33,88,40,81),(34,89,41,82),(35,90,42,83),(36,85,37,84)], [(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 | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | S3×D4 | D4⋊2S3 | Q8⋊3S3 |
| kernel | D6⋊C4⋊3C4 | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C2×C4⋊Dic3 | C2×D6⋊C4 | D6⋊C4 | C2.C42 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 8 | 1 | 2 | 2 | 3 | 8 | 4 | 4 | 4 | 1 | 2 | 1 |
Matrix representation of D6⋊C4⋊3C4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
D6⋊C4⋊3C4 in GAP, Magma, Sage, TeX
D_6\rtimes C_4\rtimes_3C_4
% in TeX
G:=Group("D6:C4:3C4"); // GroupNames label
G:=SmallGroup(192,229);
// by ID
G=gap.SmallGroup(192,229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,758,387,58,6278]);
// 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,d*b*d^-1=a^3*b*c^2,d*c*d^-1=a^3*c^-1>;
// generators/relations