direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Q8⋊3S3, C42.231D6, Q8⋊9(C4×S3), (Q8×C12)⋊8C2, (C4×Q8)⋊23S3, (C4×D12)⋊36C2, D12⋊14(C2×C4), (S3×C42)⋊6C2, C4⋊C4.324D6, C12⋊18(C4○D4), (Q8×Dic3)⋊32C2, C6.26(C23×C4), (C2×Q8).225D6, Dic3⋊5D4⋊47C2, C12.36(C22×C4), (C2×C6).117C24, D6.10(C22×C4), Dic3⋊12(C4○D4), (C2×C12).496C23, (C4×C12).169C22, D6⋊C4.125C22, C22.36(S3×C23), (C6×Q8).217C22, (C2×D12).261C22, C4⋊Dic3.367C22, Dic3.25(C22×C4), (C22×S3).176C23, (C2×Dic3).213C23, (C4×Dic3).252C22, C3⋊4(C4×C4○D4), C4.36(S3×C2×C4), (C4×S3)⋊8(C2×C4), C2.6(S3×C4○D4), C4⋊C4⋊7S3⋊47C2, (C3×Q8)⋊11(C2×C4), C2.28(S3×C22×C4), C6.111(C2×C4○D4), C2.3(C2×Q8⋊3S3), (S3×C2×C4).246C22, (C2×Q8⋊3S3).9C2, (C3×C4⋊C4).345C22, (C2×C4).821(C22×S3), SmallGroup(192,1132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Q8⋊3S3
G = < a,b,c,d,e | a4=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 664 in 310 conjugacy classes, 157 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C4×C4○D4, S3×C42, C4×D12, C4⋊C4⋊7S3, Dic3⋊5D4, Q8×Dic3, Q8×C12, C2×Q8⋊3S3, C4×Q8⋊3S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C23×C4, C2×C4○D4, S3×C2×C4, Q8⋊3S3, S3×C23, C4×C4○D4, S3×C22×C4, C2×Q8⋊3S3, S3×C4○D4, C4×Q8⋊3S3
►On 96 points
Generators in S96
(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 51 27 47)(2 52 28 48)(3 49 25 45)(4 50 26 46)(5 67 91 88)(6 68 92 85)(7 65 89 86)(8 66 90 87)(9 61 37 44)(10 62 38 41)(11 63 39 42)(12 64 40 43)(13 70 94 74)(14 71 95 75)(15 72 96 76)(16 69 93 73)(17 60 84 80)(18 57 81 77)(19 58 82 78)(20 59 83 79)(21 55 31 35)(22 56 32 36)(23 53 29 33)(24 54 30 34)
(1 73 27 69)(2 74 28 70)(3 75 25 71)(4 76 26 72)(5 62 91 41)(6 63 92 42)(7 64 89 43)(8 61 90 44)(9 87 37 66)(10 88 38 67)(11 85 39 68)(12 86 40 65)(13 52 94 48)(14 49 95 45)(15 50 96 46)(16 51 93 47)(17 54 84 34)(18 55 81 35)(19 56 82 36)(20 53 83 33)(21 77 31 57)(22 78 32 58)(23 79 29 59)(24 80 30 60)
(1 23 37)(2 24 38)(3 21 39)(4 22 40)(5 94 84)(6 95 81)(7 96 82)(8 93 83)(9 27 29)(10 28 30)(11 25 31)(12 26 32)(13 17 91)(14 18 92)(15 19 89)(16 20 90)(33 61 47)(34 62 48)(35 63 45)(36 64 46)(41 52 54)(42 49 55)(43 50 56)(44 51 53)(57 85 71)(58 86 72)(59 87 69)(60 88 70)(65 76 78)(66 73 79)(67 74 80)(68 75 77)
(1 45)(2 46)(3 47)(4 48)(5 58)(6 59)(7 60)(8 57)(9 55)(10 56)(11 53)(12 54)(13 76)(14 73)(15 74)(16 75)(17 65)(18 66)(19 67)(20 68)(21 61)(22 62)(23 63)(24 64)(25 51)(26 52)(27 49)(28 50)(29 42)(30 43)(31 44)(32 41)(33 39)(34 40)(35 37)(36 38)(69 95)(70 96)(71 93)(72 94)(77 90)(78 91)(79 92)(80 89)(81 87)(82 88)(83 85)(84 86)
G:=sub<Sym(96)| (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), (1,51,27,47)(2,52,28,48)(3,49,25,45)(4,50,26,46)(5,67,91,88)(6,68,92,85)(7,65,89,86)(8,66,90,87)(9,61,37,44)(10,62,38,41)(11,63,39,42)(12,64,40,43)(13,70,94,74)(14,71,95,75)(15,72,96,76)(16,69,93,73)(17,60,84,80)(18,57,81,77)(19,58,82,78)(20,59,83,79)(21,55,31,35)(22,56,32,36)(23,53,29,33)(24,54,30,34), (1,73,27,69)(2,74,28,70)(3,75,25,71)(4,76,26,72)(5,62,91,41)(6,63,92,42)(7,64,89,43)(8,61,90,44)(9,87,37,66)(10,88,38,67)(11,85,39,68)(12,86,40,65)(13,52,94,48)(14,49,95,45)(15,50,96,46)(16,51,93,47)(17,54,84,34)(18,55,81,35)(19,56,82,36)(20,53,83,33)(21,77,31,57)(22,78,32,58)(23,79,29,59)(24,80,30,60), (1,23,37)(2,24,38)(3,21,39)(4,22,40)(5,94,84)(6,95,81)(7,96,82)(8,93,83)(9,27,29)(10,28,30)(11,25,31)(12,26,32)(13,17,91)(14,18,92)(15,19,89)(16,20,90)(33,61,47)(34,62,48)(35,63,45)(36,64,46)(41,52,54)(42,49,55)(43,50,56)(44,51,53)(57,85,71)(58,86,72)(59,87,69)(60,88,70)(65,76,78)(66,73,79)(67,74,80)(68,75,77), (1,45)(2,46)(3,47)(4,48)(5,58)(6,59)(7,60)(8,57)(9,55)(10,56)(11,53)(12,54)(13,76)(14,73)(15,74)(16,75)(17,65)(18,66)(19,67)(20,68)(21,61)(22,62)(23,63)(24,64)(25,51)(26,52)(27,49)(28,50)(29,42)(30,43)(31,44)(32,41)(33,39)(34,40)(35,37)(36,38)(69,95)(70,96)(71,93)(72,94)(77,90)(78,91)(79,92)(80,89)(81,87)(82,88)(83,85)(84,86)>;
G:=Group( (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), (1,51,27,47)(2,52,28,48)(3,49,25,45)(4,50,26,46)(5,67,91,88)(6,68,92,85)(7,65,89,86)(8,66,90,87)(9,61,37,44)(10,62,38,41)(11,63,39,42)(12,64,40,43)(13,70,94,74)(14,71,95,75)(15,72,96,76)(16,69,93,73)(17,60,84,80)(18,57,81,77)(19,58,82,78)(20,59,83,79)(21,55,31,35)(22,56,32,36)(23,53,29,33)(24,54,30,34), (1,73,27,69)(2,74,28,70)(3,75,25,71)(4,76,26,72)(5,62,91,41)(6,63,92,42)(7,64,89,43)(8,61,90,44)(9,87,37,66)(10,88,38,67)(11,85,39,68)(12,86,40,65)(13,52,94,48)(14,49,95,45)(15,50,96,46)(16,51,93,47)(17,54,84,34)(18,55,81,35)(19,56,82,36)(20,53,83,33)(21,77,31,57)(22,78,32,58)(23,79,29,59)(24,80,30,60), (1,23,37)(2,24,38)(3,21,39)(4,22,40)(5,94,84)(6,95,81)(7,96,82)(8,93,83)(9,27,29)(10,28,30)(11,25,31)(12,26,32)(13,17,91)(14,18,92)(15,19,89)(16,20,90)(33,61,47)(34,62,48)(35,63,45)(36,64,46)(41,52,54)(42,49,55)(43,50,56)(44,51,53)(57,85,71)(58,86,72)(59,87,69)(60,88,70)(65,76,78)(66,73,79)(67,74,80)(68,75,77), (1,45)(2,46)(3,47)(4,48)(5,58)(6,59)(7,60)(8,57)(9,55)(10,56)(11,53)(12,54)(13,76)(14,73)(15,74)(16,75)(17,65)(18,66)(19,67)(20,68)(21,61)(22,62)(23,63)(24,64)(25,51)(26,52)(27,49)(28,50)(29,42)(30,43)(31,44)(32,41)(33,39)(34,40)(35,37)(36,38)(69,95)(70,96)(71,93)(72,94)(77,90)(78,91)(79,92)(80,89)(81,87)(82,88)(83,85)(84,86) );
G=PermutationGroup([[(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)], [(1,51,27,47),(2,52,28,48),(3,49,25,45),(4,50,26,46),(5,67,91,88),(6,68,92,85),(7,65,89,86),(8,66,90,87),(9,61,37,44),(10,62,38,41),(11,63,39,42),(12,64,40,43),(13,70,94,74),(14,71,95,75),(15,72,96,76),(16,69,93,73),(17,60,84,80),(18,57,81,77),(19,58,82,78),(20,59,83,79),(21,55,31,35),(22,56,32,36),(23,53,29,33),(24,54,30,34)], [(1,73,27,69),(2,74,28,70),(3,75,25,71),(4,76,26,72),(5,62,91,41),(6,63,92,42),(7,64,89,43),(8,61,90,44),(9,87,37,66),(10,88,38,67),(11,85,39,68),(12,86,40,65),(13,52,94,48),(14,49,95,45),(15,50,96,46),(16,51,93,47),(17,54,84,34),(18,55,81,35),(19,56,82,36),(20,53,83,33),(21,77,31,57),(22,78,32,58),(23,79,29,59),(24,80,30,60)], [(1,23,37),(2,24,38),(3,21,39),(4,22,40),(5,94,84),(6,95,81),(7,96,82),(8,93,83),(9,27,29),(10,28,30),(11,25,31),(12,26,32),(13,17,91),(14,18,92),(15,19,89),(16,20,90),(33,61,47),(34,62,48),(35,63,45),(36,64,46),(41,52,54),(42,49,55),(43,50,56),(44,51,53),(57,85,71),(58,86,72),(59,87,69),(60,88,70),(65,76,78),(66,73,79),(67,74,80),(68,75,77)], [(1,45),(2,46),(3,47),(4,48),(5,58),(6,59),(7,60),(8,57),(9,55),(10,56),(11,53),(12,54),(13,76),(14,73),(15,74),(16,75),(17,65),(18,66),(19,67),(20,68),(21,61),(22,62),(23,63),(24,64),(25,51),(26,52),(27,49),(28,50),(29,42),(30,43),(31,44),(32,41),(33,39),(34,40),(35,37),(36,38),(69,95),(70,96),(71,93),(72,94),(77,90),(78,91),(79,92),(80,89),(81,87),(82,88),(83,85),(84,86)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X | 4Y | ··· | 4AD | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4×S3 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C4×Q8⋊3S3 | S3×C42 | C4×D12 | C4⋊C4⋊7S3 | Dic3⋊5D4 | Q8×Dic3 | Q8×C12 | C2×Q8⋊3S3 | Q8⋊3S3 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Dic3 | C12 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C4×Q8⋊3S3 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,0,8,0,0,8,0],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C4×Q8⋊3S3 in GAP, Magma, Sage, TeX
C_4\times Q_8\rtimes_3S_3
% in TeX
G:=Group("C4xQ8:3S3"); // GroupNames label
G:=SmallGroup(192,1132);
// by ID
G=gap.SmallGroup(192,1132);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^3=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations