direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×Q8⋊6D4, C6.1622+ 1+4, Q8⋊7(C3×D4), (C4×D4)⋊18C6, (C3×Q8)⋊24D4, (C4×Q8)⋊17C6, C4.44(C6×D4), (D4×C12)⋊47C2, C4⋊D4⋊14C6, C4⋊1D4⋊10C6, (Q8×C12)⋊33C2, C12⋊19(C4○D4), C12.405(C2×D4), C42.45(C2×C6), (C2×C6).370C24, C6.198(C22×D4), (C4×C12).286C22, (C2×C12).677C23, (C6×D4).220C22, C23.17(C22×C6), C22.44(C23×C6), (C6×Q8).286C22, (C22×C6).101C23, C2.14(C3×2+ 1+4), (C22×C12).456C22, C4⋊3(C3×C4○D4), C2.22(D4×C2×C6), (C6×C4○D4)⋊25C2, (C2×C4○D4)⋊13C6, C4⋊C4.72(C2×C6), C2.23(C6×C4○D4), (C3×C4⋊D4)⋊41C2, (C3×C4⋊1D4)⋊19C2, (C2×D4).68(C2×C6), C6.242(C2×C4○D4), (C2×Q8).86(C2×C6), C22⋊C4.21(C2×C6), (C22×C4).73(C2×C6), (C3×C4⋊C4).398C22, (C2×C4).137(C22×C6), (C3×C22⋊C4).154C22, SmallGroup(192,1439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊6D4
G = < a,b,c,d,e | a3=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, Q8⋊6D4, D4×C12, Q8×C12, C3×C4⋊D4, C3×C4⋊1D4, C6×C4○D4, C3×Q8⋊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, Q8⋊6D4, D4×C2×C6, C6×C4○D4, C3×2+ 1+4, C3×Q8⋊6D4
►On 96 points
Generators in S96
(1 26 23)(2 27 24)(3 28 21)(4 25 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)(13 93 84)(14 94 81)(15 95 82)(16 96 83)(29 38 42)(30 39 43)(31 40 44)(32 37 41)(33 49 46)(34 50 47)(35 51 48)(36 52 45)(53 62 66)(54 63 67)(55 64 68)(56 61 65)(57 73 70)(58 74 71)(59 75 72)(60 76 69)(77 86 90)(78 87 91)(79 88 92)(80 85 89)
(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 33 3 35)(2 36 4 34)(5 38 7 40)(6 37 8 39)(9 42 11 44)(10 41 12 43)(13 75 15 73)(14 74 16 76)(17 30 19 32)(18 29 20 31)(21 48 23 46)(22 47 24 45)(25 50 27 52)(26 49 28 51)(53 78 55 80)(54 77 56 79)(57 84 59 82)(58 83 60 81)(61 88 63 86)(62 87 64 85)(65 92 67 90)(66 91 68 89)(69 94 71 96)(70 93 72 95)
(1 63 8 57)(2 62 5 60)(3 61 6 59)(4 64 7 58)(9 76 27 66)(10 75 28 65)(11 74 25 68)(12 73 26 67)(13 51 90 41)(14 50 91 44)(15 49 92 43)(16 52 89 42)(17 70 23 54)(18 69 24 53)(19 72 21 56)(20 71 22 55)(29 96 45 80)(30 95 46 79)(31 94 47 78)(32 93 48 77)(33 88 39 82)(34 87 40 81)(35 86 37 84)(36 85 38 83)
(1 58)(2 59)(3 60)(4 57)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 52)(14 49)(15 50)(16 51)(17 55)(18 56)(19 53)(20 54)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
G:=sub<Sym(96)| (1,26,23)(2,27,24)(3,28,21)(4,25,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,93,84)(14,94,81)(15,95,82)(16,96,83)(29,38,42)(30,39,43)(31,40,44)(32,37,41)(33,49,46)(34,50,47)(35,51,48)(36,52,45)(53,62,66)(54,63,67)(55,64,68)(56,61,65)(57,73,70)(58,74,71)(59,75,72)(60,76,69)(77,86,90)(78,87,91)(79,88,92)(80,85,89), (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), (1,33,3,35)(2,36,4,34)(5,38,7,40)(6,37,8,39)(9,42,11,44)(10,41,12,43)(13,75,15,73)(14,74,16,76)(17,30,19,32)(18,29,20,31)(21,48,23,46)(22,47,24,45)(25,50,27,52)(26,49,28,51)(53,78,55,80)(54,77,56,79)(57,84,59,82)(58,83,60,81)(61,88,63,86)(62,87,64,85)(65,92,67,90)(66,91,68,89)(69,94,71,96)(70,93,72,95), (1,63,8,57)(2,62,5,60)(3,61,6,59)(4,64,7,58)(9,76,27,66)(10,75,28,65)(11,74,25,68)(12,73,26,67)(13,51,90,41)(14,50,91,44)(15,49,92,43)(16,52,89,42)(17,70,23,54)(18,69,24,53)(19,72,21,56)(20,71,22,55)(29,96,45,80)(30,95,46,79)(31,94,47,78)(32,93,48,77)(33,88,39,82)(34,87,40,81)(35,86,37,84)(36,85,38,83), (1,58)(2,59)(3,60)(4,57)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,52)(14,49)(15,50)(16,51)(17,55)(18,56)(19,53)(20,54)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)>;
G:=Group( (1,26,23)(2,27,24)(3,28,21)(4,25,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,93,84)(14,94,81)(15,95,82)(16,96,83)(29,38,42)(30,39,43)(31,40,44)(32,37,41)(33,49,46)(34,50,47)(35,51,48)(36,52,45)(53,62,66)(54,63,67)(55,64,68)(56,61,65)(57,73,70)(58,74,71)(59,75,72)(60,76,69)(77,86,90)(78,87,91)(79,88,92)(80,85,89), (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), (1,33,3,35)(2,36,4,34)(5,38,7,40)(6,37,8,39)(9,42,11,44)(10,41,12,43)(13,75,15,73)(14,74,16,76)(17,30,19,32)(18,29,20,31)(21,48,23,46)(22,47,24,45)(25,50,27,52)(26,49,28,51)(53,78,55,80)(54,77,56,79)(57,84,59,82)(58,83,60,81)(61,88,63,86)(62,87,64,85)(65,92,67,90)(66,91,68,89)(69,94,71,96)(70,93,72,95), (1,63,8,57)(2,62,5,60)(3,61,6,59)(4,64,7,58)(9,76,27,66)(10,75,28,65)(11,74,25,68)(12,73,26,67)(13,51,90,41)(14,50,91,44)(15,49,92,43)(16,52,89,42)(17,70,23,54)(18,69,24,53)(19,72,21,56)(20,71,22,55)(29,96,45,80)(30,95,46,79)(31,94,47,78)(32,93,48,77)(33,88,39,82)(34,87,40,81)(35,86,37,84)(36,85,38,83), (1,58)(2,59)(3,60)(4,57)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,52)(14,49)(15,50)(16,51)(17,55)(18,56)(19,53)(20,54)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96) );
G=PermutationGroup([[(1,26,23),(2,27,24),(3,28,21),(4,25,22),(5,9,18),(6,10,19),(7,11,20),(8,12,17),(13,93,84),(14,94,81),(15,95,82),(16,96,83),(29,38,42),(30,39,43),(31,40,44),(32,37,41),(33,49,46),(34,50,47),(35,51,48),(36,52,45),(53,62,66),(54,63,67),(55,64,68),(56,61,65),(57,73,70),(58,74,71),(59,75,72),(60,76,69),(77,86,90),(78,87,91),(79,88,92),(80,85,89)], [(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)], [(1,33,3,35),(2,36,4,34),(5,38,7,40),(6,37,8,39),(9,42,11,44),(10,41,12,43),(13,75,15,73),(14,74,16,76),(17,30,19,32),(18,29,20,31),(21,48,23,46),(22,47,24,45),(25,50,27,52),(26,49,28,51),(53,78,55,80),(54,77,56,79),(57,84,59,82),(58,83,60,81),(61,88,63,86),(62,87,64,85),(65,92,67,90),(66,91,68,89),(69,94,71,96),(70,93,72,95)], [(1,63,8,57),(2,62,5,60),(3,61,6,59),(4,64,7,58),(9,76,27,66),(10,75,28,65),(11,74,25,68),(12,73,26,67),(13,51,90,41),(14,50,91,44),(15,49,92,43),(16,52,89,42),(17,70,23,54),(18,69,24,53),(19,72,21,56),(20,71,22,55),(29,96,45,80),(30,95,46,79),(31,94,47,78),(32,93,48,77),(33,88,39,82),(34,87,40,81),(35,86,37,84),(36,85,38,83)], [(1,58),(2,59),(3,60),(4,57),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,52),(14,49),(15,50),(16,51),(17,55),(18,56),(19,53),(20,54),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | ··· | 4L | 4M | 4N | 4O | 6A | ··· | 6F | 6G | ··· | 6R | 12A | ··· | 12X | 12Y | ··· | 12AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 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 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | 2+ 1+4 | C3×2+ 1+4 |
| kernel | C3×Q8⋊6D4 | D4×C12 | Q8×C12 | C3×C4⋊D4 | C3×C4⋊1D4 | C6×C4○D4 | Q8⋊6D4 | C4×D4 | C4×Q8 | C4⋊D4 | C4⋊1D4 | C2×C4○D4 | C3×Q8 | C12 | Q8 | C4 | C6 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 2 | 6 | 2 | 12 | 6 | 4 | 4 | 4 | 8 | 8 | 1 | 2 |
Matrix representation of C3×Q8⋊6D4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 11 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 10 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 5 | 10 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,10,8,0,0,0,0,1,0,0,0,0,1],[5,8,0,0,0,8,0,0,0,0,0,1,0,0,12,0],[5,8,0,0,10,8,0,0,0,0,0,1,0,0,1,0] >;
C3×Q8⋊6D4 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes_6D_4
% in TeX
G:=Group("C3xQ8:6D4"); // GroupNames label
G:=SmallGroup(192,1439);
// by ID
G=gap.SmallGroup(192,1439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,268,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations