metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (S3×Q8)⋊2C4, (C4×S3).5D4, C4⋊C4.148D6, C4.163(S3×D4), (C2×C8).175D6, Q8.14(C4×S3), Q8⋊C4⋊16S3, Q8⋊2Dic3⋊4C2, C12.117(C2×D4), (C2×Q8).128D6, Dic6.4(C2×C4), C22.77(S3×D4), C6.SD16⋊10C2, C12.13(C22×C4), C2.Dic12⋊28C2, C2.4(D4.D6), C2.2(Q16⋊S3), (C22×S3).74D4, (C6×Q8).25C22, (C2×C24).240C22, (C2×C12).242C23, D6.12(C22⋊C4), (C2×Dic3).152D4, C6.34(C8.C22), C3⋊1(C23.38D4), C4⋊Dic3.90C22, Dic3.9(C22⋊C4), (C2×Dic6).69C22, C4.13(S3×C2×C4), (C2×S3×Q8).3C2, (C4×S3).3(C2×C4), (C3×Q8).3(C2×C4), C4⋊C4⋊7S3.1C2, (C2×C8⋊S3).6C2, (C2×C6).255(C2×D4), C2.22(S3×C22⋊C4), C6.21(C2×C22⋊C4), (C2×C3⋊C8).35C22, (S3×C2×C4).17C22, (C3×Q8⋊C4)⋊23C2, (C3×C4⋊C4).43C22, (C2×C4).349(C22×S3), SmallGroup(192,361)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S3×Q8)⋊C4
G = < a,b,c,d,e | a3=b2=c4=e4=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc2, dcd-1=ece-1=c-1, ede-1=cd >
Subgroups: 392 in 150 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, Q8⋊C4, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C8⋊S3, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, S3×Q8, C6×Q8, C23.38D4, C6.SD16, C2.Dic12, Q8⋊2Dic3, C3×Q8⋊C4, C4⋊C4⋊7S3, C2×C8⋊S3, C2×S3×Q8, (S3×Q8)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C8.C22, S3×C2×C4, S3×D4, C23.38D4, S3×C22⋊C4, D4.D6, Q16⋊S3, (S3×Q8)⋊C4
►On 96 points
Generators in S96
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)(25 32)(26 29)(27 30)(28 31)(37 42)(38 43)(39 44)(40 41)(49 54)(50 55)(51 56)(52 53)(57 59)(58 60)(61 68)(62 65)(63 66)(64 67)(69 71)(70 72)(73 80)(74 77)(75 78)(76 79)(85 90)(86 91)(87 92)(88 89)
(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 60 3 58)(2 59 4 57)(5 56 7 54)(6 55 8 53)(9 52 11 50)(10 51 12 49)(13 66 15 68)(14 65 16 67)(17 62 19 64)(18 61 20 63)(21 69 23 71)(22 72 24 70)(25 78 27 80)(26 77 28 79)(29 74 31 76)(30 73 32 75)(33 81 35 83)(34 84 36 82)(37 90 39 92)(38 89 40 91)(41 86 43 88)(42 85 44 87)(45 93 47 95)(46 96 48 94)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 76 85 67)(6 75 86 66)(7 74 87 65)(8 73 88 68)(9 78 89 63)(10 77 90 62)(11 80 91 61)(12 79 92 64)(13 56 32 44)(14 55 29 43)(15 54 30 42)(16 53 31 41)(17 52 26 40)(18 51 27 39)(19 50 28 38)(20 49 25 37)(57 94 69 82)(58 93 70 81)(59 96 71 84)(60 95 72 83)
G:=sub<Sym(96)| (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(49,54)(50,55)(51,56)(52,53)(57,59)(58,60)(61,68)(62,65)(63,66)(64,67)(69,71)(70,72)(73,80)(74,77)(75,78)(76,79)(85,90)(86,91)(87,92)(88,89), (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), (1,60,3,58)(2,59,4,57)(5,56,7,54)(6,55,8,53)(9,52,11,50)(10,51,12,49)(13,66,15,68)(14,65,16,67)(17,62,19,64)(18,61,20,63)(21,69,23,71)(22,72,24,70)(25,78,27,80)(26,77,28,79)(29,74,31,76)(30,73,32,75)(33,81,35,83)(34,84,36,82)(37,90,39,92)(38,89,40,91)(41,86,43,88)(42,85,44,87)(45,93,47,95)(46,96,48,94), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83)>;
G:=Group( (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(49,54)(50,55)(51,56)(52,53)(57,59)(58,60)(61,68)(62,65)(63,66)(64,67)(69,71)(70,72)(73,80)(74,77)(75,78)(76,79)(85,90)(86,91)(87,92)(88,89), (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), (1,60,3,58)(2,59,4,57)(5,56,7,54)(6,55,8,53)(9,52,11,50)(10,51,12,49)(13,66,15,68)(14,65,16,67)(17,62,19,64)(18,61,20,63)(21,69,23,71)(22,72,24,70)(25,78,27,80)(26,77,28,79)(29,74,31,76)(30,73,32,75)(33,81,35,83)(34,84,36,82)(37,90,39,92)(38,89,40,91)(41,86,43,88)(42,85,44,87)(45,93,47,95)(46,96,48,94), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83) );
G=PermutationGroup([[(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,95,10),(6,96,11),(7,93,12),(8,94,9),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,39,44),(34,40,41),(35,37,42),(36,38,43),(45,51,56),(46,52,53),(47,49,54),(48,50,55),(57,63,68),(58,64,65),(59,61,66),(60,62,67),(69,78,73),(70,79,74),(71,80,75),(72,77,76),(81,92,87),(82,89,88),(83,90,85),(84,91,86)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24),(25,32),(26,29),(27,30),(28,31),(37,42),(38,43),(39,44),(40,41),(49,54),(50,55),(51,56),(52,53),(57,59),(58,60),(61,68),(62,65),(63,66),(64,67),(69,71),(70,72),(73,80),(74,77),(75,78),(76,79),(85,90),(86,91),(87,92),(88,89)], [(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)], [(1,60,3,58),(2,59,4,57),(5,56,7,54),(6,55,8,53),(9,52,11,50),(10,51,12,49),(13,66,15,68),(14,65,16,67),(17,62,19,64),(18,61,20,63),(21,69,23,71),(22,72,24,70),(25,78,27,80),(26,77,28,79),(29,74,31,76),(30,73,32,75),(33,81,35,83),(34,84,36,82),(37,90,39,92),(38,89,40,91),(41,86,43,88),(42,85,44,87),(45,93,47,95),(46,96,48,94)], [(1,46,22,34),(2,45,23,33),(3,48,24,36),(4,47,21,35),(5,76,85,67),(6,75,86,66),(7,74,87,65),(8,73,88,68),(9,78,89,63),(10,77,90,62),(11,80,91,61),(12,79,92,64),(13,56,32,44),(14,55,29,43),(15,54,30,42),(16,53,31,41),(17,52,26,40),(18,51,27,39),(19,50,28,38),(20,49,25,37),(57,94,69,82),(58,93,70,81),(59,96,71,84),(60,95,72,83)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4×S3 | C8.C22 | S3×D4 | S3×D4 | D4.D6 | Q16⋊S3 |
| kernel | (S3×Q8)⋊C4 | C6.SD16 | C2.Dic12 | Q8⋊2Dic3 | C3×Q8⋊C4 | C4⋊C4⋊7S3 | C2×C8⋊S3 | C2×S3×Q8 | S3×Q8 | Q8⋊C4 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of (S3×Q8)⋊C4 ►in GL8(𝔽73)
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 54 | 1 | 70 |
| 0 | 0 | 0 | 0 | 6 | 12 | 25 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 29 | 33 | 57 |
| 0 | 0 | 0 | 0 | 39 | 2 | 59 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 19 | 18 | 71 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 2 | 62 | 24 | 55 |
G:=sub<GL(8,GF(73))| [72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,72,12,0,0,0,0,0,72,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,55,6,0,0,0,0,71,72,54,12,0,0,0,0,0,0,1,25,0,0,0,0,0,0,70,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,57,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,0,40,19,17,39,0,0,0,0,31,33,29,2,0,0,0,0,0,0,33,59,0,0,0,0,0,0,57,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,19,1,2,0,0,0,0,0,18,0,62,0,0,0,0,71,71,72,24,0,0,0,0,0,3,0,55] >;
(S3×Q8)⋊C4 in GAP, Magma, Sage, TeX
(S_3\times Q_8)\rtimes C_4
% in TeX
G:=Group("(S3xQ8):C4"); // GroupNames label
G:=SmallGroup(192,361);
// by ID
G=gap.SmallGroup(192,361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^2,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c*d>;
// generators/relations