direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3⋊C4, C23.34D6, C22.4Dic6, C6⋊1(C4⋊C4), (C2×C6).5Q8, C6.7(C2×Q8), C6.38(C2×D4), (C2×C6).35D4, (C2×C4).64D6, (C2×Dic3)⋊4C4, Dic3⋊4(C2×C4), (C22×C4).5S3, C2.2(C2×Dic6), C6.17(C22×C4), (C22×C12).4C2, C22.16(C4×S3), (C2×C6).41C23, (C2×C12).76C22, C22.19(C3⋊D4), (C22×C6).33C22, C22.20(C22×S3), (C22×Dic3).4C2, (C2×Dic3).33C22, C3⋊2(C2×C4⋊C4), C2.18(S3×C2×C4), C2.1(C2×C3⋊D4), (C2×C6).17(C2×C4), SmallGroup(96,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic3⋊C4
G = < a,b,c,d | a2=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >
Subgroups: 162 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, Dic3⋊C4, C22×Dic3, C22×C12, C2×Dic3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×Dic3⋊C4
►Regular action on 96 points
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(55 76)(56 77)(57 78)(58 73)(59 74)(60 75)(61 94)(62 95)(63 96)(64 91)(65 92)(66 93)(67 88)(68 89)(69 90)(70 85)(71 86)(72 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 56 4 59)(2 55 5 58)(3 60 6 57)(7 50 10 53)(8 49 11 52)(9 54 12 51)(13 68 16 71)(14 67 17 70)(15 72 18 69)(19 62 22 65)(20 61 23 64)(21 66 24 63)(25 80 28 83)(26 79 29 82)(27 84 30 81)(31 74 34 77)(32 73 35 76)(33 78 36 75)(37 92 40 95)(38 91 41 94)(39 96 42 93)(43 86 46 89)(44 85 47 88)(45 90 48 87)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 70 55 64)(50 71 56 65)(51 72 57 66)(52 67 58 61)(53 68 59 62)(54 69 60 63)(73 94 79 88)(74 95 80 89)(75 96 81 90)(76 91 82 85)(77 92 83 86)(78 93 84 87)
G:=sub<Sym(96)| (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,94)(62,95)(63,96)(64,91)(65,92)(66,93)(67,88)(68,89)(69,90)(70,85)(71,86)(72,87), (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), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,50,10,53)(8,49,11,52)(9,54,12,51)(13,68,16,71)(14,67,17,70)(15,72,18,69)(19,62,22,65)(20,61,23,64)(21,66,24,63)(25,80,28,83)(26,79,29,82)(27,84,30,81)(31,74,34,77)(32,73,35,76)(33,78,36,75)(37,92,40,95)(38,91,41,94)(39,96,42,93)(43,86,46,89)(44,85,47,88)(45,90,48,87), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,70,55,64)(50,71,56,65)(51,72,57,66)(52,67,58,61)(53,68,59,62)(54,69,60,63)(73,94,79,88)(74,95,80,89)(75,96,81,90)(76,91,82,85)(77,92,83,86)(78,93,84,87)>;
G:=Group( (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,94)(62,95)(63,96)(64,91)(65,92)(66,93)(67,88)(68,89)(69,90)(70,85)(71,86)(72,87), (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), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,50,10,53)(8,49,11,52)(9,54,12,51)(13,68,16,71)(14,67,17,70)(15,72,18,69)(19,62,22,65)(20,61,23,64)(21,66,24,63)(25,80,28,83)(26,79,29,82)(27,84,30,81)(31,74,34,77)(32,73,35,76)(33,78,36,75)(37,92,40,95)(38,91,41,94)(39,96,42,93)(43,86,46,89)(44,85,47,88)(45,90,48,87), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,70,55,64)(50,71,56,65)(51,72,57,66)(52,67,58,61)(53,68,59,62)(54,69,60,63)(73,94,79,88)(74,95,80,89)(75,96,81,90)(76,91,82,85)(77,92,83,86)(78,93,84,87) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,28),(8,29),(9,30),(10,25),(11,26),(12,27),(13,46),(14,47),(15,48),(16,43),(17,44),(18,45),(19,40),(20,41),(21,42),(22,37),(23,38),(24,39),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(55,76),(56,77),(57,78),(58,73),(59,74),(60,75),(61,94),(62,95),(63,96),(64,91),(65,92),(66,93),(67,88),(68,89),(69,90),(70,85),(71,86),(72,87)], [(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)], [(1,56,4,59),(2,55,5,58),(3,60,6,57),(7,50,10,53),(8,49,11,52),(9,54,12,51),(13,68,16,71),(14,67,17,70),(15,72,18,69),(19,62,22,65),(20,61,23,64),(21,66,24,63),(25,80,28,83),(26,79,29,82),(27,84,30,81),(31,74,34,77),(32,73,35,76),(33,78,36,75),(37,92,40,95),(38,91,41,94),(39,96,42,93),(43,86,46,89),(44,85,47,88),(45,90,48,87)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,43,31,37),(26,44,32,38),(27,45,33,39),(28,46,34,40),(29,47,35,41),(30,48,36,42),(49,70,55,64),(50,71,56,65),(51,72,57,66),(52,67,58,61),(53,68,59,62),(54,69,60,63),(73,94,79,88),(74,95,80,89),(75,96,81,90),(76,91,82,85),(77,92,83,86),(78,93,84,87)]])
C2×Dic3⋊C4 is a maximal subgroup of
(C2×Dic3)⋊C8 (C2×C12)⋊Q8 C6.(C4×Q8) Dic3⋊C42 C3⋊(C42⋊8C4) C6.(C4×D4) C2.(C4×D12) C2.(C4×Dic6) Dic3⋊C4⋊C4 (C2×C4)⋊Dic6 C6.(C4⋊Q8) (C2×Dic3).9D4 (C2×C4).17D12 (C2×C4).Dic6 (C22×C4).85D6 D6⋊(C4⋊C4) D6⋊C4⋊C4 D6⋊C4⋊5C4 C6.C22≀C2 C6.(C4⋊D4) (C22×C4).37D6 C12⋊4(C4⋊C4) (C2×C42).6S3 (C2×C42)⋊3S3 C24.55D6 C24.14D6 C24.15D6 C24.57D6 C24.17D6 C24.18D6 C24.20D6 C24.24D6 C24.25D6 C12⋊(C4⋊C4) (C4×Dic3)⋊8C4 Dic3⋊(C4⋊C4) C6.67(C4×D4) (C2×Dic3)⋊Q8 (C2×C4).44D12 (C2×C12).54D4 (C2×Dic3).Q8 (C2×C12).288D4 D6⋊C4⋊6C4 (C2×C12).290D4 (C2×C12).56D4 C24.73D6 C24.31D6 C22.52(S3×Q8) C2×C4×Dic6 C2×S3×C4⋊C4 C42.96D6 D4⋊5Dic6 C42.104D6 C42.108D6 C42.118D6 C6.322+ 1+4 C6.342+ 1+4 C6.702- 1+4 C6.752- 1+4 C6.522+ 1+4 C6.782- 1+4 C6.802- 1+4 C6.822- 1+4 C2×C4×C3⋊D4 C6.1042- 1+4
C2×Dic3⋊C4 is a maximal quotient of
C12⋊4(C4⋊C4) C24.55D6 C24.57D6 C4⋊C4.225D6 C12⋊(C4⋊C4) (C4×Dic3)⋊8C4 (C4×Dic3)⋊9C4 C4⋊C4.232D6 C4⋊C4.234D6 Dic3⋊C8⋊C2 Dic3⋊4M4(2) C12.88(C2×Q8) C23.8Dic6 C24.73D6
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 |
| kernel | C2×Dic3⋊C4 | Dic3⋊C4 | C22×Dic3 | C22×C12 | C2×Dic3 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C2×Dic3⋊C4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 4 | 2 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,1,0],[1,0,0,0,0,0,10,10,0,0,0,7,3,0,0,0,0,0,8,0,0,0,0,8,5],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,11,4,0,0,0,9,2] >;
C2×Dic3⋊C4 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C2xDic3:C4"); // GroupNames label
G:=SmallGroup(96,130);
// by ID
G=gap.SmallGroup(96,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,362,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations