metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊5C4, C6.24(C4×D4), C6.5(C4⋊D4), C22.63(S3×D4), C2.4(Dic3⋊D4), C2.C42⋊4S3, C6.C42⋊3C2, (C2×Dic3).84D4, (C22×C4).315D6, C2.6(Dic3⋊5D4), C6.6(C42⋊C2), C2.4(D6.D4), C6.18(C4.4D4), (S3×C23).1C22, C6.20(C42⋊2C2), C2.10(C42⋊2S3), C2.9(Dic3⋊4D4), C22.37(C4○D12), (C22×C12).14C22, (C22×C6).294C23, C23.269(C22×S3), C3⋊1(C24.C22), C22.38(D4⋊2S3), C2.3(C23.11D6), C22.19(Q8⋊3S3), C6.39(C22.D4), (C22×Dic3).17C22, (C2×C4).60(C4×S3), (C2×D6⋊C4).4C2, (C2×C4×Dic3)⋊17C2, C22.92(S3×C2×C4), (C2×Dic3⋊C4)⋊2C2, (C2×C6).203(C2×D4), C2.3(C4⋊C4⋊S3), (C2×C12).144(C2×C4), (C22×S3).7(C2×C4), (C2×C6).53(C22×C4), (C2×Dic3).9(C2×C4), (C2×C6).133(C4○D4), (C3×C2.C42)⋊21C2, SmallGroup(192,228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C4⋊5C4
G = < a,b,c,d | a6=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd-1=bc2, dcd-1=a3c >
Subgroups: 544 in 190 conjugacy classes, 67 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, Dic3⋊C4, D6⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C24.C22, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4, D6⋊C4⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C24.C22, C42⋊2S3, Dic3⋊4D4, Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C4⋊C4⋊S3, D6⋊C4⋊5C4
►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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 24)(10 23)(11 22)(12 21)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)(49 65)(50 64)(51 63)(52 62)(53 61)(54 66)(55 71)(56 70)(57 69)(58 68)(59 67)(60 72)(73 95)(74 94)(75 93)(76 92)(77 91)(78 96)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 71 11 65)(2 72 12 66)(3 67 7 61)(4 68 8 62)(5 69 9 63)(6 70 10 64)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 79 43 73)(38 80 44 74)(39 81 45 75)(40 82 46 76)(41 83 47 77)(42 84 48 78)
(1 41 17 29)(2 42 18 30)(3 37 13 25)(4 38 14 26)(5 39 15 27)(6 40 16 28)(7 43 19 31)(8 44 20 32)(9 45 21 33)(10 46 22 34)(11 47 23 35)(12 48 24 36)(49 88 61 76)(50 89 62 77)(51 90 63 78)(52 85 64 73)(53 86 65 74)(54 87 66 75)(55 94 67 82)(56 95 68 83)(57 96 69 84)(58 91 70 79)(59 92 71 80)(60 93 72 81)
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,24)(10,23)(11,22)(12,21)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39)(49,65)(50,64)(51,63)(52,62)(53,61)(54,66)(55,71)(56,70)(57,69)(58,68)(59,67)(60,72)(73,95)(74,94)(75,93)(76,92)(77,91)(78,96)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,71,11,65)(2,72,12,66)(3,67,7,61)(4,68,8,62)(5,69,9,63)(6,70,10,64)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,79,43,73)(38,80,44,74)(39,81,45,75)(40,82,46,76)(41,83,47,77)(42,84,48,78), (1,41,17,29)(2,42,18,30)(3,37,13,25)(4,38,14,26)(5,39,15,27)(6,40,16,28)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,85,64,73)(53,86,65,74)(54,87,66,75)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81)>;
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,24)(10,23)(11,22)(12,21)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,38)(32,37)(33,42)(34,41)(35,40)(36,39)(49,65)(50,64)(51,63)(52,62)(53,61)(54,66)(55,71)(56,70)(57,69)(58,68)(59,67)(60,72)(73,95)(74,94)(75,93)(76,92)(77,91)(78,96)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,71,11,65)(2,72,12,66)(3,67,7,61)(4,68,8,62)(5,69,9,63)(6,70,10,64)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,79,43,73)(38,80,44,74)(39,81,45,75)(40,82,46,76)(41,83,47,77)(42,84,48,78), (1,41,17,29)(2,42,18,30)(3,37,13,25)(4,38,14,26)(5,39,15,27)(6,40,16,28)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,85,64,73)(53,86,65,74)(54,87,66,75)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,20),(8,19),(9,24),(10,23),(11,22),(12,21),(25,44),(26,43),(27,48),(28,47),(29,46),(30,45),(31,38),(32,37),(33,42),(34,41),(35,40),(36,39),(49,65),(50,64),(51,63),(52,62),(53,61),(54,66),(55,71),(56,70),(57,69),(58,68),(59,67),(60,72),(73,95),(74,94),(75,93),(76,92),(77,91),(78,96),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90)], [(1,71,11,65),(2,72,12,66),(3,67,7,61),(4,68,8,62),(5,69,9,63),(6,70,10,64),(13,55,19,49),(14,56,20,50),(15,57,21,51),(16,58,22,52),(17,59,23,53),(18,60,24,54),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,79,43,73),(38,80,44,74),(39,81,45,75),(40,82,46,76),(41,83,47,77),(42,84,48,78)], [(1,41,17,29),(2,42,18,30),(3,37,13,25),(4,38,14,26),(5,39,15,27),(6,40,16,28),(7,43,19,31),(8,44,20,32),(9,45,21,33),(10,46,22,34),(11,47,23,35),(12,48,24,36),(49,88,61,76),(50,89,62,77),(51,90,63,78),(52,85,64,73),(53,86,65,74),(54,87,66,75),(55,94,67,82),(56,95,68,83),(57,96,69,84),(58,91,70,79),(59,92,71,80),(60,93,72,81)]])
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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | C4○D12 | S3×D4 | D4⋊2S3 | Q8⋊3S3 |
| kernel | D6⋊C4⋊5C4 | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×D6⋊C4 | D6⋊C4 | C2.C42 | C2×Dic3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 8 | 1 | 4 | 3 | 8 | 4 | 8 | 2 | 1 | 1 |
Matrix representation of D6⋊C4⋊5C4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 1 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,5,0,0,0,0,0,1],[8,10,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,1,0,0,0,0,2,8],[12,2,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,5,0,0,0,0,0,1] >;
D6⋊C4⋊5C4 in GAP, Magma, Sage, TeX
D_6\rtimes C_4\rtimes_5C_4
% in TeX
G:=Group("D6:C4:5C4"); // GroupNames label
G:=SmallGroup(192,228);
// by ID
G=gap.SmallGroup(192,228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,422,387,58,6278]);
// 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=b*c^2,d*c*d^-1=a^3*c>;
// generators/relations