direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C3⋊Q16, C6⋊2Q16, C12.20D4, Q8.12D6, C12.16C23, Dic6.10C22, C3⋊3(C2×Q16), C6.55(C2×D4), (C2×C4).54D6, (C2×C6).43D4, C3⋊C8.9C22, (C6×Q8).3C2, (C2×Q8).5S3, C4.9(C3⋊D4), C4.16(C22×S3), (C2×Dic6).8C2, (C3×Q8).7C22, (C2×C12).38C22, C22.24(C3⋊D4), (C2×C3⋊C8).6C2, C2.19(C2×C3⋊D4), SmallGroup(96,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊Q16
G = < a,b,c,d | a2=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 114 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C2×Q16, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, C2×C3⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C2×Q16, C3⋊Q16, C2×C3⋊D4, C2×C3⋊Q16
Character table of C2×C3⋊Q16
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ19 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -√-3 | -√-3 | √-3 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | √-3 | √-3 | -√-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
►Regular action on 96 points
Generators in S96
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 81)(64 82)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 17 91)(2 92 18)(3 19 93)(4 94 20)(5 21 95)(6 96 22)(7 23 89)(8 90 24)(9 39 72)(10 65 40)(11 33 66)(12 67 34)(13 35 68)(14 69 36)(15 37 70)(16 71 38)(25 57 73)(26 74 58)(27 59 75)(28 76 60)(29 61 77)(30 78 62)(31 63 79)(32 80 64)(41 55 87)(42 88 56)(43 49 81)(44 82 50)(45 51 83)(46 84 52)(47 53 85)(48 86 54)
(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 42 13 46)(10 41 14 45)(11 48 15 44)(12 47 16 43)(17 64 21 60)(18 63 22 59)(19 62 23 58)(20 61 24 57)(25 94 29 90)(26 93 30 89)(27 92 31 96)(28 91 32 95)(33 86 37 82)(34 85 38 81)(35 84 39 88)(36 83 40 87)(49 67 53 71)(50 66 54 70)(51 65 55 69)(52 72 56 68)
G:=sub<Sym(96)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,17,91)(2,92,18)(3,19,93)(4,94,20)(5,21,95)(6,96,22)(7,23,89)(8,90,24)(9,39,72)(10,65,40)(11,33,66)(12,67,34)(13,35,68)(14,69,36)(15,37,70)(16,71,38)(25,57,73)(26,74,58)(27,59,75)(28,76,60)(29,61,77)(30,78,62)(31,63,79)(32,80,64)(41,55,87)(42,88,56)(43,49,81)(44,82,50)(45,51,83)(46,84,52)(47,53,85)(48,86,54), (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)(9,42,13,46)(10,41,14,45)(11,48,15,44)(12,47,16,43)(17,64,21,60)(18,63,22,59)(19,62,23,58)(20,61,24,57)(25,94,29,90)(26,93,30,89)(27,92,31,96)(28,91,32,95)(33,86,37,82)(34,85,38,81)(35,84,39,88)(36,83,40,87)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(65,94)(66,95)(67,96)(68,89)(69,90)(70,91)(71,92)(72,93), (1,17,91)(2,92,18)(3,19,93)(4,94,20)(5,21,95)(6,96,22)(7,23,89)(8,90,24)(9,39,72)(10,65,40)(11,33,66)(12,67,34)(13,35,68)(14,69,36)(15,37,70)(16,71,38)(25,57,73)(26,74,58)(27,59,75)(28,76,60)(29,61,77)(30,78,62)(31,63,79)(32,80,64)(41,55,87)(42,88,56)(43,49,81)(44,82,50)(45,51,83)(46,84,52)(47,53,85)(48,86,54), (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)(9,42,13,46)(10,41,14,45)(11,48,15,44)(12,47,16,43)(17,64,21,60)(18,63,22,59)(19,62,23,58)(20,61,24,57)(25,94,29,90)(26,93,30,89)(27,92,31,96)(28,91,32,95)(33,86,37,82)(34,85,38,81)(35,84,39,88)(36,83,40,87)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68) );
G=PermutationGroup([[(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,81),(64,82),(65,94),(66,95),(67,96),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,17,91),(2,92,18),(3,19,93),(4,94,20),(5,21,95),(6,96,22),(7,23,89),(8,90,24),(9,39,72),(10,65,40),(11,33,66),(12,67,34),(13,35,68),(14,69,36),(15,37,70),(16,71,38),(25,57,73),(26,74,58),(27,59,75),(28,76,60),(29,61,77),(30,78,62),(31,63,79),(32,80,64),(41,55,87),(42,88,56),(43,49,81),(44,82,50),(45,51,83),(46,84,52),(47,53,85),(48,86,54)], [(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),(9,42,13,46),(10,41,14,45),(11,48,15,44),(12,47,16,43),(17,64,21,60),(18,63,22,59),(19,62,23,58),(20,61,24,57),(25,94,29,90),(26,93,30,89),(27,92,31,96),(28,91,32,95),(33,86,37,82),(34,85,38,81),(35,84,39,88),(36,83,40,87),(49,67,53,71),(50,66,54,70),(51,65,55,69),(52,72,56,68)]])
C2×C3⋊Q16 is a maximal subgroup of
D12.7D4 C3⋊Q16⋊C4 Dic3⋊4Q16 Dic3⋊Q16 Dic6.11D4 Q8.11D12 D6⋊Q16 D6⋊1Q16 C3⋊C8.D4 Q8.6D12 C42.59D6 C12⋊7Q16 D12.37D4 Dic6.37D4 C3⋊C8.29D4 C3⋊C8.6D4 C42.61D6 C42.214D6 C42.65D6 C42.80D6 C12⋊Q16 C12⋊3Q16 (C3×Q8).D4 C24.31D4 C24.43D4 Dic6.16D4 Dic3⋊3Q16 C24.26D4 D6⋊5Q16 C24.37D4 M4(2).16D6 (C2×C6)⋊8Q16 (C3×D4).32D4 C2×S3×Q16 SD16.D6 D12.35C23
C2×C3⋊Q16 is a maximal quotient of
C4⋊C4.230D6 Q8⋊5Dic6 C12⋊7Q16 (C2×C6).Q16 Dic6.37D4 C3⋊C8.29D4 C12.17D8 C12.9Q16 C12⋊Q16 Dic6⋊5Q8 C12⋊3Q16 C12.Q16 (C2×C6)⋊8Q16
Matrix representation of C2×C3⋊Q16 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 26 | 39 |
| 0 | 0 | 65 | 47 |
| 12 | 72 | 0 | 0 |
| 72 | 61 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[16,16,0,0,57,16,0,0,0,0,26,65,0,0,39,47],[12,72,0,0,72,61,0,0,0,0,43,13,0,0,60,30] >;
C2×C3⋊Q16 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes Q_{16} % in TeX
G:=Group("C2xC3:Q16"); // GroupNames label
G:=SmallGroup(96,150);
// by ID
G=gap.SmallGroup(96,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,86,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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
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