direct product, metabelian, supersoluble, monomial
Aliases: C6×C3⋊Q16, C62.126D4, (C3×C6)⋊6Q16, C3⋊3(C6×Q16), C6⋊2(C3×Q16), C6.55(C6×D4), (C3×C12).90D4, C12.20(C3×D4), (C3×Q8).72D6, (C6×Q8).11C6, (C6×Q8).28S3, Q8.17(S3×C6), C32⋊13(C2×Q16), (C2×C12).330D6, (C2×Dic6).8C6, C12.91(C3⋊D4), C12.16(C22×C6), (C3×C12).87C23, (C6×Dic6).14C2, Dic6.10(C2×C6), (C6×C12).126C22, C12.167(C22×S3), (C3×Dic6).40C22, (Q8×C32).24C22, (C2×C3⋊C8).6C6, C3⋊C8.9(C2×C6), C4.16(S3×C2×C6), (C6×C3⋊C8).14C2, (Q8×C3×C6).5C2, C4.9(C3×C3⋊D4), (C2×C4).54(S3×C6), (C2×C6).52(C3×D4), C2.19(C6×C3⋊D4), (C2×Q8).7(C3×S3), (C2×C12).37(C2×C6), (C3×C6).263(C2×D4), C6.156(C2×C3⋊D4), (C3×C3⋊C8).36C22, (C3×Q8).19(C2×C6), C22.24(C3×C3⋊D4), (C2×C6).117(C3⋊D4), SmallGroup(288,714)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C3⋊Q16
G = < a,b,c,d | a6=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 266 in 139 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C6×Q8, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C6×C12, C6×C12, Q8×C32, Q8×C32, C2×C3⋊Q16, C6×Q16, C6×C3⋊C8, C3×C3⋊Q16, C6×Dic6, Q8×C3×C6, C6×C3⋊Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, Q16, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C2×Q16, S3×C6, C3⋊Q16, C3×Q16, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C2×C3⋊Q16, C6×Q16, C3×C3⋊Q16, C6×C3⋊D4, C6×C3⋊Q16
►On 96 points
(1 70 55 61 91 45)(2 71 56 62 92 46)(3 72 49 63 93 47)(4 65 50 64 94 48)(5 66 51 57 95 41)(6 67 52 58 96 42)(7 68 53 59 89 43)(8 69 54 60 90 44)(9 19 39 86 76 32)(10 20 40 87 77 25)(11 21 33 88 78 26)(12 22 34 81 79 27)(13 23 35 82 80 28)(14 24 36 83 73 29)(15 17 37 84 74 30)(16 18 38 85 75 31)
(1 55 91)(2 92 56)(3 49 93)(4 94 50)(5 51 95)(6 96 52)(7 53 89)(8 90 54)(9 39 76)(10 77 40)(11 33 78)(12 79 34)(13 35 80)(14 73 36)(15 37 74)(16 75 38)(17 84 30)(18 31 85)(19 86 32)(20 25 87)(21 88 26)(22 27 81)(23 82 28)(24 29 83)(41 66 57)(42 58 67)(43 68 59)(44 60 69)(45 70 61)(46 62 71)(47 72 63)(48 64 65)
(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 80 5 76)(2 79 6 75)(3 78 7 74)(4 77 8 73)(9 55 13 51)(10 54 14 50)(11 53 15 49)(12 52 16 56)(17 63 21 59)(18 62 22 58)(19 61 23 57)(20 60 24 64)(25 69 29 65)(26 68 30 72)(27 67 31 71)(28 66 32 70)(33 89 37 93)(34 96 38 92)(35 95 39 91)(36 94 40 90)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)
G:=sub<Sym(96)| (1,70,55,61,91,45)(2,71,56,62,92,46)(3,72,49,63,93,47)(4,65,50,64,94,48)(5,66,51,57,95,41)(6,67,52,58,96,42)(7,68,53,59,89,43)(8,69,54,60,90,44)(9,19,39,86,76,32)(10,20,40,87,77,25)(11,21,33,88,78,26)(12,22,34,81,79,27)(13,23,35,82,80,28)(14,24,36,83,73,29)(15,17,37,84,74,30)(16,18,38,85,75,31), (1,55,91)(2,92,56)(3,49,93)(4,94,50)(5,51,95)(6,96,52)(7,53,89)(8,90,54)(9,39,76)(10,77,40)(11,33,78)(12,79,34)(13,35,80)(14,73,36)(15,37,74)(16,75,38)(17,84,30)(18,31,85)(19,86,32)(20,25,87)(21,88,26)(22,27,81)(23,82,28)(24,29,83)(41,66,57)(42,58,67)(43,68,59)(44,60,69)(45,70,61)(46,62,71)(47,72,63)(48,64,65), (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), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,60,24,64)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,89,37,93)(34,96,38,92)(35,95,39,91)(36,94,40,90)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)>;
G:=Group( (1,70,55,61,91,45)(2,71,56,62,92,46)(3,72,49,63,93,47)(4,65,50,64,94,48)(5,66,51,57,95,41)(6,67,52,58,96,42)(7,68,53,59,89,43)(8,69,54,60,90,44)(9,19,39,86,76,32)(10,20,40,87,77,25)(11,21,33,88,78,26)(12,22,34,81,79,27)(13,23,35,82,80,28)(14,24,36,83,73,29)(15,17,37,84,74,30)(16,18,38,85,75,31), (1,55,91)(2,92,56)(3,49,93)(4,94,50)(5,51,95)(6,96,52)(7,53,89)(8,90,54)(9,39,76)(10,77,40)(11,33,78)(12,79,34)(13,35,80)(14,73,36)(15,37,74)(16,75,38)(17,84,30)(18,31,85)(19,86,32)(20,25,87)(21,88,26)(22,27,81)(23,82,28)(24,29,83)(41,66,57)(42,58,67)(43,68,59)(44,60,69)(45,70,61)(46,62,71)(47,72,63)(48,64,65), (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), (1,80,5,76)(2,79,6,75)(3,78,7,74)(4,77,8,73)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,60,24,64)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,89,37,93)(34,96,38,92)(35,95,39,91)(36,94,40,90)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87) );
G=PermutationGroup([[(1,70,55,61,91,45),(2,71,56,62,92,46),(3,72,49,63,93,47),(4,65,50,64,94,48),(5,66,51,57,95,41),(6,67,52,58,96,42),(7,68,53,59,89,43),(8,69,54,60,90,44),(9,19,39,86,76,32),(10,20,40,87,77,25),(11,21,33,88,78,26),(12,22,34,81,79,27),(13,23,35,82,80,28),(14,24,36,83,73,29),(15,17,37,84,74,30),(16,18,38,85,75,31)], [(1,55,91),(2,92,56),(3,49,93),(4,94,50),(5,51,95),(6,96,52),(7,53,89),(8,90,54),(9,39,76),(10,77,40),(11,33,78),(12,79,34),(13,35,80),(14,73,36),(15,37,74),(16,75,38),(17,84,30),(18,31,85),(19,86,32),(20,25,87),(21,88,26),(22,27,81),(23,82,28),(24,29,83),(41,66,57),(42,58,67),(43,68,59),(44,60,69),(45,70,61),(46,62,71),(47,72,63),(48,64,65)], [(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)], [(1,80,5,76),(2,79,6,75),(3,78,7,74),(4,77,8,73),(9,55,13,51),(10,54,14,50),(11,53,15,49),(12,52,16,56),(17,63,21,59),(18,62,22,58),(19,61,23,57),(20,60,24,64),(25,69,29,65),(26,68,30,72),(27,67,31,71),(28,66,32,70),(33,89,37,93),(34,96,38,92),(35,95,39,91),(36,94,40,90),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×Q16 | C3×C3⋊D4 | C3×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 |
| kernel | C6×C3⋊Q16 | C6×C3⋊C8 | C3×C3⋊Q16 | C6×Dic6 | Q8×C3×C6 | C2×C3⋊Q16 | C2×C3⋊C8 | C3⋊Q16 | C2×Dic6 | C6×Q8 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C3×C6 | C2×Q8 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | Q8 | C6 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 4 |
Matrix representation of C6×C3⋊Q16 ►in GL5(𝔽73)
| 65 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 64 | 9 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 44 | 0 | 0 |
| 0 | 63 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 |
| 0 | 0 | 0 | 39 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 44 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 32 | 49 |
| 0 | 0 | 0 | 64 | 41 |
G:=sub<GL(5,GF(73))| [65,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,64,0,0,0,0,9,8,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,63,0,0,0,44,1,0,0,0,0,0,0,39,0,0,0,58,32],[1,0,0,0,0,0,72,0,0,0,0,44,1,0,0,0,0,0,32,64,0,0,0,49,41] >;
C6×C3⋊Q16 in GAP, Magma, Sage, TeX
C_6\times C_3\rtimes Q_{16} % in TeX
G:=Group("C6xC3:Q16"); // GroupNames label
G:=SmallGroup(288,714);
// by ID
G=gap.SmallGroup(288,714);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,268,2524,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^8=1,d^2=c^4,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=c^-1>;
// generators/relations