direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×Q8⋊C4, Q8⋊2C12, C6.6Q16, C12.61D4, C6.10SD16, C4⋊C4.1C6, (C2×C8).1C6, (C3×Q8)⋊4C4, (C2×C24).3C2, C4.2(C2×C12), (C2×C6).47D4, C4.12(C3×D4), (C2×Q8).4C6, (C6×Q8).7C2, C2.1(C3×Q16), C12.29(C2×C4), C2.2(C3×SD16), C22.9(C3×D4), C6.25(C22⋊C4), (C2×C12).115C22, (C3×C4⋊C4).8C2, (C2×C4).18(C2×C6), C2.7(C3×C22⋊C4), SmallGroup(96,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊C4
G = < a,b,c,d | a3=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Smallest permutation representation of C3×Q8⋊C4
►Regular action on 96 points
Generators in S96
(1 22 14)(2 23 15)(3 24 16)(4 21 13)(5 91 83)(6 92 84)(7 89 81)(8 90 82)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(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 34 3 36)(2 33 4 35)(5 69 7 71)(6 72 8 70)(9 30 11 32)(10 29 12 31)(13 43 15 41)(14 42 16 44)(17 38 19 40)(18 37 20 39)(21 51 23 49)(22 50 24 52)(25 45 27 47)(26 48 28 46)(53 81 55 83)(54 84 56 82)(57 78 59 80)(58 77 60 79)(61 89 63 91)(62 92 64 90)(65 86 67 88)(66 85 68 87)(73 95 75 93)(74 94 76 96)
(1 60 11 54)(2 59 12 53)(3 58 9 56)(4 57 10 55)(5 52 94 46)(6 51 95 45)(7 50 96 48)(8 49 93 47)(13 65 18 63)(14 68 19 62)(15 67 20 61)(16 66 17 64)(21 74 25 71)(22 73 26 70)(23 76 27 69)(24 75 28 72)(29 84 35 79)(30 83 36 78)(31 82 33 77)(32 81 34 80)(37 92 43 87)(38 91 44 86)(39 90 41 85)(40 89 42 88)
G:=sub<Sym(96)| (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,34,3,36)(2,33,4,35)(5,69,7,71)(6,72,8,70)(9,30,11,32)(10,29,12,31)(13,43,15,41)(14,42,16,44)(17,38,19,40)(18,37,20,39)(21,51,23,49)(22,50,24,52)(25,45,27,47)(26,48,28,46)(53,81,55,83)(54,84,56,82)(57,78,59,80)(58,77,60,79)(61,89,63,91)(62,92,64,90)(65,86,67,88)(66,85,68,87)(73,95,75,93)(74,94,76,96), (1,60,11,54)(2,59,12,53)(3,58,9,56)(4,57,10,55)(5,52,94,46)(6,51,95,45)(7,50,96,48)(8,49,93,47)(13,65,18,63)(14,68,19,62)(15,67,20,61)(16,66,17,64)(21,74,25,71)(22,73,26,70)(23,76,27,69)(24,75,28,72)(29,84,35,79)(30,83,36,78)(31,82,33,77)(32,81,34,80)(37,92,43,87)(38,91,44,86)(39,90,41,85)(40,89,42,88)>;
G:=Group( (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,34,3,36)(2,33,4,35)(5,69,7,71)(6,72,8,70)(9,30,11,32)(10,29,12,31)(13,43,15,41)(14,42,16,44)(17,38,19,40)(18,37,20,39)(21,51,23,49)(22,50,24,52)(25,45,27,47)(26,48,28,46)(53,81,55,83)(54,84,56,82)(57,78,59,80)(58,77,60,79)(61,89,63,91)(62,92,64,90)(65,86,67,88)(66,85,68,87)(73,95,75,93)(74,94,76,96), (1,60,11,54)(2,59,12,53)(3,58,9,56)(4,57,10,55)(5,52,94,46)(6,51,95,45)(7,50,96,48)(8,49,93,47)(13,65,18,63)(14,68,19,62)(15,67,20,61)(16,66,17,64)(21,74,25,71)(22,73,26,70)(23,76,27,69)(24,75,28,72)(29,84,35,79)(30,83,36,78)(31,82,33,77)(32,81,34,80)(37,92,43,87)(38,91,44,86)(39,90,41,85)(40,89,42,88) );
G=PermutationGroup([[(1,22,14),(2,23,15),(3,24,16),(4,21,13),(5,91,83),(6,92,84),(7,89,81),(8,90,82),(9,28,17),(10,25,18),(11,26,19),(12,27,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,49,41),(34,50,42),(35,51,43),(36,52,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,74,65),(58,75,66),(59,76,67),(60,73,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(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)], [(1,34,3,36),(2,33,4,35),(5,69,7,71),(6,72,8,70),(9,30,11,32),(10,29,12,31),(13,43,15,41),(14,42,16,44),(17,38,19,40),(18,37,20,39),(21,51,23,49),(22,50,24,52),(25,45,27,47),(26,48,28,46),(53,81,55,83),(54,84,56,82),(57,78,59,80),(58,77,60,79),(61,89,63,91),(62,92,64,90),(65,86,67,88),(66,85,68,87),(73,95,75,93),(74,94,76,96)], [(1,60,11,54),(2,59,12,53),(3,58,9,56),(4,57,10,55),(5,52,94,46),(6,51,95,45),(7,50,96,48),(8,49,93,47),(13,65,18,63),(14,68,19,62),(15,67,20,61),(16,66,17,64),(21,74,25,71),(22,73,26,70),(23,76,27,69),(24,75,28,72),(29,84,35,79),(30,83,36,78),(31,82,33,77),(32,81,34,80),(37,92,43,87),(38,91,44,86),(39,90,41,85),(40,89,42,88)]])
C3×Q8⋊C4 is a maximal subgroup of
Dic3⋊7SD16 C3⋊Q16⋊C4 Dic3⋊4Q16 Q8⋊2Dic6 Dic3.1Q16 Q8⋊3Dic6 (C2×C8).D6 Dic3⋊Q16 Q8.3Dic6 (C2×Q8).36D6 Dic6.11D4 Q8.4Dic6 Q8⋊C4⋊S3 (S3×Q8)⋊C4 Q8⋊7(C4×S3) C4⋊C4.150D6 D6.1SD16 Q8⋊3D12 D6⋊2SD16 Q8.11D12 D6⋊Q16 Q8⋊4D12 D6.Q16 C3⋊(C8⋊D4) D6⋊1Q16 D6⋊C8.C2 C8⋊Dic3⋊C2 C3⋊C8.D4 Q8⋊3(C4×S3) Dic3⋊SD16 D12.12D4 C12×SD16 C12×Q16
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | SD16 | Q16 | C3×D4 | C3×D4 | C3×SD16 | C3×Q16 |
| kernel | C3×Q8⋊C4 | C3×C4⋊C4 | C2×C24 | C6×Q8 | Q8⋊C4 | C3×Q8 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C12 | C2×C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C3×Q8⋊C4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 72 | 0 |
| 1 | 0 | 0 |
| 0 | 65 | 9 |
| 0 | 9 | 8 |
| 46 | 0 | 0 |
| 0 | 6 | 29 |
| 0 | 29 | 67 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[1,0,0,0,0,72,0,1,0],[1,0,0,0,65,9,0,9,8],[46,0,0,0,6,29,0,29,67] >;
C3×Q8⋊C4 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes C_4
% in TeX
G:=Group("C3xQ8:C4"); // GroupNames label
G:=SmallGroup(96,53);
// by ID
G=gap.SmallGroup(96,53);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295,1443,729,117]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations
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