metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6)⋊8Q16, (C3×Q8).31D4, C6.47(C2×Q16), C6.75C22≀C2, (C2×C12).304D4, C12.211(C2×D4), (C2×Q8).193D6, C3⋊5(C22⋊Q16), Q8⋊2Dic3⋊39C2, Q8.20(C3⋊D4), C22⋊4(C3⋊Q16), (C22×C6).201D4, (C22×C4).173D6, (C22×Q8).10S3, (C2×C12).478C23, C2.9(C24⋊4S3), C23.96(C3⋊D4), (C6×Q8).204C22, C12.55D4.10C2, C12.48D4.14C2, C6.103(C8.C22), C4⋊Dic3.188C22, C2.23(Q8.11D6), (C22×C12).204C22, (C2×Dic6).138C22, (Q8×C2×C6).4C2, C4.61(C2×C3⋊D4), (C2×C3⋊Q16)⋊23C2, (C2×C6).561(C2×D4), C2.17(C2×C3⋊Q16), (C2×C4).87(C3⋊D4), (C2×C3⋊C8).176C22, (C2×C4).563(C22×S3), C22.221(C2×C3⋊D4), SmallGroup(192,787)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6)⋊8Q16
G = < a,b,c,d | a2=b6=c8=1, d2=c4, ab=ba, cac-1=ab3, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 328 in 148 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C6.D4, C2×Dic6, C22×C12, C22×C12, C6×Q8, C6×Q8, C22⋊Q16, C12.55D4, Q8⋊2Dic3, C12.48D4, C2×C3⋊Q16, Q8×C2×C6, (C2×C6)⋊8Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16, C2×C3⋊D4, C22⋊Q16, Q8.11D6, C2×C3⋊Q16, C24⋊4S3, (C2×C6)⋊8Q16
►On 96 points
(1 5)(2 25)(3 7)(4 27)(6 29)(8 31)(9 52)(10 14)(11 54)(12 16)(13 56)(15 50)(17 21)(18 41)(19 23)(20 43)(22 45)(24 47)(26 30)(28 32)(33 68)(34 38)(35 70)(36 40)(37 72)(39 66)(42 46)(44 48)(49 53)(51 55)(57 94)(58 62)(59 96)(60 64)(61 90)(63 92)(65 69)(67 71)(73 77)(74 87)(75 79)(76 81)(78 83)(80 85)(82 86)(84 88)(89 93)(91 95)
(1 14 79 28 53 88)(2 81 54 29 80 15)(3 16 73 30 55 82)(4 83 56 31 74 9)(5 10 75 32 49 84)(6 85 50 25 76 11)(7 12 77 26 51 86)(8 87 52 27 78 13)(17 67 64 44 36 89)(18 90 37 45 57 68)(19 69 58 46 38 91)(20 92 39 47 59 70)(21 71 60 48 40 93)(22 94 33 41 61 72)(23 65 62 42 34 95)(24 96 35 43 63 66)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 70 13 66)(10 69 14 65)(11 68 15 72)(12 67 16 71)(25 45 29 41)(26 44 30 48)(27 43 31 47)(28 42 32 46)(33 50 37 54)(34 49 38 53)(35 56 39 52)(36 55 40 51)(57 80 61 76)(58 79 62 75)(59 78 63 74)(60 77 64 73)(81 94 85 90)(82 93 86 89)(83 92 87 96)(84 91 88 95)
G:=sub<Sym(96)| (1,5)(2,25)(3,7)(4,27)(6,29)(8,31)(9,52)(10,14)(11,54)(12,16)(13,56)(15,50)(17,21)(18,41)(19,23)(20,43)(22,45)(24,47)(26,30)(28,32)(33,68)(34,38)(35,70)(36,40)(37,72)(39,66)(42,46)(44,48)(49,53)(51,55)(57,94)(58,62)(59,96)(60,64)(61,90)(63,92)(65,69)(67,71)(73,77)(74,87)(75,79)(76,81)(78,83)(80,85)(82,86)(84,88)(89,93)(91,95), (1,14,79,28,53,88)(2,81,54,29,80,15)(3,16,73,30,55,82)(4,83,56,31,74,9)(5,10,75,32,49,84)(6,85,50,25,76,11)(7,12,77,26,51,86)(8,87,52,27,78,13)(17,67,64,44,36,89)(18,90,37,45,57,68)(19,69,58,46,38,91)(20,92,39,47,59,70)(21,71,60,48,40,93)(22,94,33,41,61,72)(23,65,62,42,34,95)(24,96,35,43,63,66), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(81,94,85,90)(82,93,86,89)(83,92,87,96)(84,91,88,95)>;
G:=Group( (1,5)(2,25)(3,7)(4,27)(6,29)(8,31)(9,52)(10,14)(11,54)(12,16)(13,56)(15,50)(17,21)(18,41)(19,23)(20,43)(22,45)(24,47)(26,30)(28,32)(33,68)(34,38)(35,70)(36,40)(37,72)(39,66)(42,46)(44,48)(49,53)(51,55)(57,94)(58,62)(59,96)(60,64)(61,90)(63,92)(65,69)(67,71)(73,77)(74,87)(75,79)(76,81)(78,83)(80,85)(82,86)(84,88)(89,93)(91,95), (1,14,79,28,53,88)(2,81,54,29,80,15)(3,16,73,30,55,82)(4,83,56,31,74,9)(5,10,75,32,49,84)(6,85,50,25,76,11)(7,12,77,26,51,86)(8,87,52,27,78,13)(17,67,64,44,36,89)(18,90,37,45,57,68)(19,69,58,46,38,91)(20,92,39,47,59,70)(21,71,60,48,40,93)(22,94,33,41,61,72)(23,65,62,42,34,95)(24,96,35,43,63,66), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(81,94,85,90)(82,93,86,89)(83,92,87,96)(84,91,88,95) );
G=PermutationGroup([[(1,5),(2,25),(3,7),(4,27),(6,29),(8,31),(9,52),(10,14),(11,54),(12,16),(13,56),(15,50),(17,21),(18,41),(19,23),(20,43),(22,45),(24,47),(26,30),(28,32),(33,68),(34,38),(35,70),(36,40),(37,72),(39,66),(42,46),(44,48),(49,53),(51,55),(57,94),(58,62),(59,96),(60,64),(61,90),(63,92),(65,69),(67,71),(73,77),(74,87),(75,79),(76,81),(78,83),(80,85),(82,86),(84,88),(89,93),(91,95)], [(1,14,79,28,53,88),(2,81,54,29,80,15),(3,16,73,30,55,82),(4,83,56,31,74,9),(5,10,75,32,49,84),(6,85,50,25,76,11),(7,12,77,26,51,86),(8,87,52,27,78,13),(17,67,64,44,36,89),(18,90,37,45,57,68),(19,69,58,46,38,91),(20,92,39,47,59,70),(21,71,60,48,40,93),(22,94,33,41,61,72),(23,65,62,42,34,95),(24,96,35,43,63,66)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,70,13,66),(10,69,14,65),(11,68,15,72),(12,67,16,71),(25,45,29,41),(26,44,30,48),(27,43,31,47),(28,42,32,46),(33,50,37,54),(34,49,38,53),(35,56,39,52),(36,55,40,51),(57,80,61,76),(58,79,62,75),(59,78,63,74),(60,77,64,73),(81,94,85,90),(82,93,86,89),(83,92,87,96),(84,91,88,95)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | Q16 | C3⋊D4 | C3⋊D4 | C3⋊D4 | C8.C22 | C3⋊Q16 | Q8.11D6 |
| kernel | (C2×C6)⋊8Q16 | C12.55D4 | Q8⋊2Dic3 | C12.48D4 | C2×C3⋊Q16 | Q8×C2×C6 | C22×Q8 | C2×C12 | C3×Q8 | C22×C6 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | Q8 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 1 | 2 | 2 |
Matrix representation of (C2×C6)⋊8Q16 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 9 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 61 | 1 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,65,0,0,0,0,9],[57,57,0,0,16,57,0,0,0,0,0,72,0,0,1,0],[61,1,0,0,1,12,0,0,0,0,1,0,0,0,0,72] >;
(C2×C6)⋊8Q16 in GAP, Magma, Sage, TeX
(C_2\times C_6)\rtimes_8Q_{16} % in TeX
G:=Group("(C2xC6):8Q16"); // GroupNames label
G:=SmallGroup(192,787);
// by ID
G=gap.SmallGroup(192,787);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations