direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D6⋊C4, D6⋊2C42, (C2×C42)⋊2S3, C6.17(C4×D4), C2.3(C4×D12), C12⋊5(C22⋊C4), (C2×C4).168D12, (C2×C12).494D4, C2.13(S3×C42), C6.12(C2×C42), (C22×C4).475D6, C22.38(C2×D12), C6.C42⋊48C2, C2.4(C42⋊2S3), C6.14(C42⋊C2), C22.47(C4○D12), (S3×C23).86C22, (C22×C6).313C23, C23.281(C22×S3), (C22×C12).477C22, (C22×Dic3).181C22, (S3×C2×C4)⋊12C4, (C2×C4×C12)⋊15C2, C3⋊2(C4×C22⋊C4), (C2×C4)⋊11(C4×S3), C2.1(C2×D6⋊C4), C2.2(C4×C3⋊D4), (C2×C12)⋊27(C2×C4), (C2×C4×Dic3)⋊19C2, C22.53(S3×C2×C4), (C2×D6⋊C4).28C2, (C2×C6).427(C2×D4), C6.32(C2×C22⋊C4), (S3×C22×C4).18C2, (C2×Dic3)⋊13(C2×C4), (C2×C6).72(C4○D4), (C2×C6).99(C22×C4), C22.42(C2×C3⋊D4), (C2×C4).272(C3⋊D4), (C22×S3).40(C2×C4), SmallGroup(192,497)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D6⋊C4
G = < a,b,c,d | a4=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 632 in 258 conjugacy classes, 107 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊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×C42, C2×C22⋊C4, C23×C4, C4×Dic3, D6⋊C4, C4×C12, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C4×C22⋊C4, C6.C42, C2×C4×Dic3, C2×D6⋊C4, C2×C4×C12, S3×C22×C4, C4×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, D6⋊C4, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C4×C22⋊C4, S3×C42, C42⋊2S3, C4×D12, C2×D6⋊C4, C4×C3⋊D4, C4×D6⋊C4
►On 96 points
(1 56 11 50)(2 57 12 51)(3 58 7 52)(4 59 8 53)(5 60 9 54)(6 55 10 49)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 67 22 61)(17 68 23 62)(18 69 24 63)(25 82 31 76)(26 83 32 77)(27 84 33 78)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 94 43 88)(38 95 44 89)(39 96 45 90)(40 91 46 85)(41 92 47 86)(42 93 48 87)
(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 22)(2 21)(3 20)(4 19)(5 24)(6 23)(7 14)(8 13)(9 18)(10 17)(11 16)(12 15)(25 47)(26 46)(27 45)(28 44)(29 43)(30 48)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)(49 68)(50 67)(51 72)(52 71)(53 70)(54 69)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(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 95 23 77)(2 96 24 78)(3 91 19 73)(4 92 20 74)(5 93 21 75)(6 94 22 76)(7 85 13 79)(8 86 14 80)(9 87 15 81)(10 88 16 82)(11 89 17 83)(12 90 18 84)(25 55 43 61)(26 56 44 62)(27 57 45 63)(28 58 46 64)(29 59 47 65)(30 60 48 66)(31 49 37 67)(32 50 38 68)(33 51 39 69)(34 52 40 70)(35 53 41 71)(36 54 42 72)
G:=sub<Sym(96)| (1,56,11,50)(2,57,12,51)(3,58,7,52)(4,59,8,53)(5,60,9,54)(6,55,10,49)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,67,22,61)(17,68,23,62)(18,69,24,63)(25,82,31,76)(26,83,32,77)(27,84,33,78)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,94,43,88)(38,95,44,89)(39,96,45,90)(40,91,46,85)(41,92,47,86)(42,93,48,87), (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), (1,22)(2,21)(3,20)(4,19)(5,24)(6,23)(7,14)(8,13)(9,18)(10,17)(11,16)(12,15)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)(49,68)(50,67)(51,72)(52,71)(53,70)(54,69)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(73,95)(74,94)(75,93)(76,92)(77,91)(78,96)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,95,23,77)(2,96,24,78)(3,91,19,73)(4,92,20,74)(5,93,21,75)(6,94,22,76)(7,85,13,79)(8,86,14,80)(9,87,15,81)(10,88,16,82)(11,89,17,83)(12,90,18,84)(25,55,43,61)(26,56,44,62)(27,57,45,63)(28,58,46,64)(29,59,47,65)(30,60,48,66)(31,49,37,67)(32,50,38,68)(33,51,39,69)(34,52,40,70)(35,53,41,71)(36,54,42,72)>;
G:=Group( (1,56,11,50)(2,57,12,51)(3,58,7,52)(4,59,8,53)(5,60,9,54)(6,55,10,49)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,67,22,61)(17,68,23,62)(18,69,24,63)(25,82,31,76)(26,83,32,77)(27,84,33,78)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,94,43,88)(38,95,44,89)(39,96,45,90)(40,91,46,85)(41,92,47,86)(42,93,48,87), (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), (1,22)(2,21)(3,20)(4,19)(5,24)(6,23)(7,14)(8,13)(9,18)(10,17)(11,16)(12,15)(25,47)(26,46)(27,45)(28,44)(29,43)(30,48)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)(49,68)(50,67)(51,72)(52,71)(53,70)(54,69)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(73,95)(74,94)(75,93)(76,92)(77,91)(78,96)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,95,23,77)(2,96,24,78)(3,91,19,73)(4,92,20,74)(5,93,21,75)(6,94,22,76)(7,85,13,79)(8,86,14,80)(9,87,15,81)(10,88,16,82)(11,89,17,83)(12,90,18,84)(25,55,43,61)(26,56,44,62)(27,57,45,63)(28,58,46,64)(29,59,47,65)(30,60,48,66)(31,49,37,67)(32,50,38,68)(33,51,39,69)(34,52,40,70)(35,53,41,71)(36,54,42,72) );
G=PermutationGroup([[(1,56,11,50),(2,57,12,51),(3,58,7,52),(4,59,8,53),(5,60,9,54),(6,55,10,49),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,67,22,61),(17,68,23,62),(18,69,24,63),(25,82,31,76),(26,83,32,77),(27,84,33,78),(28,79,34,73),(29,80,35,74),(30,81,36,75),(37,94,43,88),(38,95,44,89),(39,96,45,90),(40,91,46,85),(41,92,47,86),(42,93,48,87)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,24),(6,23),(7,14),(8,13),(9,18),(10,17),(11,16),(12,15),(25,47),(26,46),(27,45),(28,44),(29,43),(30,48),(31,41),(32,40),(33,39),(34,38),(35,37),(36,42),(49,68),(50,67),(51,72),(52,71),(53,70),(54,69),(55,62),(56,61),(57,66),(58,65),(59,64),(60,63),(73,95),(74,94),(75,93),(76,92),(77,91),(78,96),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90)], [(1,95,23,77),(2,96,24,78),(3,91,19,73),(4,92,20,74),(5,93,21,75),(6,94,22,76),(7,85,13,79),(8,86,14,80),(9,87,15,81),(10,88,16,82),(11,89,17,83),(12,90,18,84),(25,55,43,61),(26,56,44,62),(27,57,45,63),(28,58,46,64),(29,59,47,65),(30,60,48,66),(31,49,37,67),(32,50,38,68),(33,51,39,69),(34,52,40,70),(35,53,41,71),(36,54,42,72)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | C4○D12 |
| kernel | C4×D6⋊C4 | C6.C42 | C2×C4×Dic3 | C2×D6⋊C4 | C2×C4×C12 | S3×C22×C4 | D6⋊C4 | S3×C2×C4 | C2×C42 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 16 | 8 | 1 | 4 | 3 | 4 | 12 | 4 | 4 | 8 |
Matrix representation of C4×D6⋊C4 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,1],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[12,0,0,0,0,8,0,0,0,0,3,7,0,0,6,10] >;
C4×D6⋊C4 in GAP, Magma, Sage, TeX
C_4\times D_6\rtimes C_4
% in TeX
G:=Group("C4xD6:C4"); // GroupNames label
G:=SmallGroup(192,497);
// by ID
G=gap.SmallGroup(192,497);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations