direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×D4⋊6D4, C6.1172- 1+4, D4⋊6(C3×D4), C4⋊Q8⋊13C6, (C3×D4)⋊24D4, (C4×D4)⋊16C6, C4.41(C6×D4), (D4×C12)⋊45C2, C4⋊D4⋊12C6, C22⋊Q8⋊12C6, C12⋊16(C4○D4), C22.6(C6×D4), C12.402(C2×D4), C42.42(C2×C6), (C2×C6).367C24, C6.195(C22×D4), C22.D4⋊9C6, (C2×C12).675C23, (C4×C12).283C22, (C6×D4).321C22, C22.41(C23×C6), C23.45(C22×C6), (C22×C6).99C23, (C6×Q8).273C22, C2.9(C3×2- 1+4), (C22×C12).453C22, C4⋊2(C3×C4○D4), (C2×C4⋊C4)⋊20C6, (C6×C4⋊C4)⋊47C2, C2.19(D4×C2×C6), (C3×C4⋊Q8)⋊34C2, (C2×C4○D4)⋊11C6, (C6×C4○D4)⋊23C2, C4⋊C4.31(C2×C6), C2.21(C6×C4○D4), (C3×C4⋊D4)⋊39C2, (C2×D4).66(C2×C6), C6.240(C2×C4○D4), (C2×C6).183(C2×D4), C22⋊C4.5(C2×C6), (C3×C22⋊Q8)⋊39C2, (C2×Q8).72(C2×C6), (C2×C4).33(C22×C6), (C22×C4).70(C2×C6), (C3×C4⋊C4).395C22, (C3×C22.D4)⋊28C2, (C3×C22⋊C4).87C22, SmallGroup(192,1436)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊6D4
G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 426 in 292 conjugacy classes, 166 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, D4⋊6D4, C6×C4⋊C4, D4×C12, C3×C4⋊D4, C3×C22⋊Q8, C3×C22.D4, C3×C4⋊Q8, C6×C4○D4, C3×D4⋊6D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C24, C3×D4, C22×C6, C22×D4, C2×C4○D4, 2- 1+4, C6×D4, C3×C4○D4, C23×C6, D4⋊6D4, D4×C2×C6, C6×C4○D4, C3×2- 1+4, C3×D4⋊6D4
►On 96 points
(1 11 51)(2 12 52)(3 9 49)(4 10 50)(5 13 65)(6 14 66)(7 15 67)(8 16 68)(17 22 54)(18 23 55)(19 24 56)(20 21 53)(25 81 45)(26 82 46)(27 83 47)(28 84 48)(29 85 89)(30 86 90)(31 87 91)(32 88 92)(33 64 93)(34 61 94)(35 62 95)(36 63 96)(37 69 57)(38 70 58)(39 71 59)(40 72 60)(41 73 77)(42 74 78)(43 75 79)(44 76 80)
(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 74)(2 73)(3 76)(4 75)(5 88)(6 87)(7 86)(8 85)(9 80)(10 79)(11 78)(12 77)(13 92)(14 91)(15 90)(16 89)(17 27)(18 26)(19 25)(20 28)(21 84)(22 83)(23 82)(24 81)(29 68)(30 67)(31 66)(32 65)(33 72)(34 71)(35 70)(36 69)(37 96)(38 95)(39 94)(40 93)(41 52)(42 51)(43 50)(44 49)(45 56)(46 55)(47 54)(48 53)(57 63)(58 62)(59 61)(60 64)
(1 34 47 6)(2 35 48 7)(3 36 45 8)(4 33 46 5)(9 63 25 16)(10 64 26 13)(11 61 27 14)(12 62 28 15)(17 91 78 59)(18 92 79 60)(19 89 80 57)(20 90 77 58)(21 30 41 38)(22 31 42 39)(23 32 43 40)(24 29 44 37)(49 96 81 68)(50 93 82 65)(51 94 83 66)(52 95 84 67)(53 86 73 70)(54 87 74 71)(55 88 75 72)(56 85 76 69)
(1 47)(2 48)(3 45)(4 46)(9 25)(10 26)(11 27)(12 28)(17 80)(18 77)(19 78)(20 79)(21 43)(22 44)(23 41)(24 42)(29 31)(30 32)(37 39)(38 40)(49 81)(50 82)(51 83)(52 84)(53 75)(54 76)(55 73)(56 74)(57 59)(58 60)(69 71)(70 72)(85 87)(86 88)(89 91)(90 92)
G:=sub<Sym(96)| (1,11,51)(2,12,52)(3,9,49)(4,10,50)(5,13,65)(6,14,66)(7,15,67)(8,16,68)(17,22,54)(18,23,55)(19,24,56)(20,21,53)(25,81,45)(26,82,46)(27,83,47)(28,84,48)(29,85,89)(30,86,90)(31,87,91)(32,88,92)(33,64,93)(34,61,94)(35,62,95)(36,63,96)(37,69,57)(38,70,58)(39,71,59)(40,72,60)(41,73,77)(42,74,78)(43,75,79)(44,76,80), (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), (1,74)(2,73)(3,76)(4,75)(5,88)(6,87)(7,86)(8,85)(9,80)(10,79)(11,78)(12,77)(13,92)(14,91)(15,90)(16,89)(17,27)(18,26)(19,25)(20,28)(21,84)(22,83)(23,82)(24,81)(29,68)(30,67)(31,66)(32,65)(33,72)(34,71)(35,70)(36,69)(37,96)(38,95)(39,94)(40,93)(41,52)(42,51)(43,50)(44,49)(45,56)(46,55)(47,54)(48,53)(57,63)(58,62)(59,61)(60,64), (1,34,47,6)(2,35,48,7)(3,36,45,8)(4,33,46,5)(9,63,25,16)(10,64,26,13)(11,61,27,14)(12,62,28,15)(17,91,78,59)(18,92,79,60)(19,89,80,57)(20,90,77,58)(21,30,41,38)(22,31,42,39)(23,32,43,40)(24,29,44,37)(49,96,81,68)(50,93,82,65)(51,94,83,66)(52,95,84,67)(53,86,73,70)(54,87,74,71)(55,88,75,72)(56,85,76,69), (1,47)(2,48)(3,45)(4,46)(9,25)(10,26)(11,27)(12,28)(17,80)(18,77)(19,78)(20,79)(21,43)(22,44)(23,41)(24,42)(29,31)(30,32)(37,39)(38,40)(49,81)(50,82)(51,83)(52,84)(53,75)(54,76)(55,73)(56,74)(57,59)(58,60)(69,71)(70,72)(85,87)(86,88)(89,91)(90,92)>;
G:=Group( (1,11,51)(2,12,52)(3,9,49)(4,10,50)(5,13,65)(6,14,66)(7,15,67)(8,16,68)(17,22,54)(18,23,55)(19,24,56)(20,21,53)(25,81,45)(26,82,46)(27,83,47)(28,84,48)(29,85,89)(30,86,90)(31,87,91)(32,88,92)(33,64,93)(34,61,94)(35,62,95)(36,63,96)(37,69,57)(38,70,58)(39,71,59)(40,72,60)(41,73,77)(42,74,78)(43,75,79)(44,76,80), (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), (1,74)(2,73)(3,76)(4,75)(5,88)(6,87)(7,86)(8,85)(9,80)(10,79)(11,78)(12,77)(13,92)(14,91)(15,90)(16,89)(17,27)(18,26)(19,25)(20,28)(21,84)(22,83)(23,82)(24,81)(29,68)(30,67)(31,66)(32,65)(33,72)(34,71)(35,70)(36,69)(37,96)(38,95)(39,94)(40,93)(41,52)(42,51)(43,50)(44,49)(45,56)(46,55)(47,54)(48,53)(57,63)(58,62)(59,61)(60,64), (1,34,47,6)(2,35,48,7)(3,36,45,8)(4,33,46,5)(9,63,25,16)(10,64,26,13)(11,61,27,14)(12,62,28,15)(17,91,78,59)(18,92,79,60)(19,89,80,57)(20,90,77,58)(21,30,41,38)(22,31,42,39)(23,32,43,40)(24,29,44,37)(49,96,81,68)(50,93,82,65)(51,94,83,66)(52,95,84,67)(53,86,73,70)(54,87,74,71)(55,88,75,72)(56,85,76,69), (1,47)(2,48)(3,45)(4,46)(9,25)(10,26)(11,27)(12,28)(17,80)(18,77)(19,78)(20,79)(21,43)(22,44)(23,41)(24,42)(29,31)(30,32)(37,39)(38,40)(49,81)(50,82)(51,83)(52,84)(53,75)(54,76)(55,73)(56,74)(57,59)(58,60)(69,71)(70,72)(85,87)(86,88)(89,91)(90,92) );
G=PermutationGroup([[(1,11,51),(2,12,52),(3,9,49),(4,10,50),(5,13,65),(6,14,66),(7,15,67),(8,16,68),(17,22,54),(18,23,55),(19,24,56),(20,21,53),(25,81,45),(26,82,46),(27,83,47),(28,84,48),(29,85,89),(30,86,90),(31,87,91),(32,88,92),(33,64,93),(34,61,94),(35,62,95),(36,63,96),(37,69,57),(38,70,58),(39,71,59),(40,72,60),(41,73,77),(42,74,78),(43,75,79),(44,76,80)], [(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)], [(1,74),(2,73),(3,76),(4,75),(5,88),(6,87),(7,86),(8,85),(9,80),(10,79),(11,78),(12,77),(13,92),(14,91),(15,90),(16,89),(17,27),(18,26),(19,25),(20,28),(21,84),(22,83),(23,82),(24,81),(29,68),(30,67),(31,66),(32,65),(33,72),(34,71),(35,70),(36,69),(37,96),(38,95),(39,94),(40,93),(41,52),(42,51),(43,50),(44,49),(45,56),(46,55),(47,54),(48,53),(57,63),(58,62),(59,61),(60,64)], [(1,34,47,6),(2,35,48,7),(3,36,45,8),(4,33,46,5),(9,63,25,16),(10,64,26,13),(11,61,27,14),(12,62,28,15),(17,91,78,59),(18,92,79,60),(19,89,80,57),(20,90,77,58),(21,30,41,38),(22,31,42,39),(23,32,43,40),(24,29,44,37),(49,96,81,68),(50,93,82,65),(51,94,83,66),(52,95,84,67),(53,86,73,70),(54,87,74,71),(55,88,75,72),(56,85,76,69)], [(1,47),(2,48),(3,45),(4,46),(9,25),(10,26),(11,27),(12,28),(17,80),(18,77),(19,78),(20,79),(21,43),(22,44),(23,41),(24,42),(29,31),(30,32),(37,39),(38,40),(49,81),(50,82),(51,83),(52,84),(53,75),(54,76),(55,73),(56,74),(57,59),(58,60),(69,71),(70,72),(85,87),(86,88),(89,91),(90,92)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4O | 6A | ··· | 6F | 6G | ··· | 6N | 6O | 6P | 6Q | 6R | 12A | ··· | 12P | 12Q | ··· | 12AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | 2- 1+4 | C3×2- 1+4 |
| kernel | C3×D4⋊6D4 | C6×C4⋊C4 | D4×C12 | C3×C4⋊D4 | C3×C22⋊Q8 | C3×C22.D4 | C3×C4⋊Q8 | C6×C4○D4 | D4⋊6D4 | C2×C4⋊C4 | C4×D4 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4⋊Q8 | C2×C4○D4 | C3×D4 | C12 | D4 | C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 2 |
Matrix representation of C3×D4⋊6D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 8 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 8 | 5 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,8,8,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,8,8,0,0,10,5],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,1,0,0,0,0,12,12,0,0,0,1] >;
C3×D4⋊6D4 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_6D_4
% in TeX
G:=Group("C3xD4:6D4"); // GroupNames label
G:=SmallGroup(192,1436);
// by ID
G=gap.SmallGroup(192,1436);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,2102,268,794]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations