metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.22D6, Q8⋊C4⋊1S3, (C2×C8).207D6, (C2×Q8).38D6, (C8×Dic3)⋊22C2, C6.68(C4○D8), Q8⋊2Dic3⋊2C2, C4.Dic6⋊5C2, C6.D8.3C2, C4.31(C4○D12), C12.18(C4○D4), (C2×Dic3).92D4, C2.D24.11C2, C22.194(S3×D4), (C6×Q8).23C22, C2.7(D24⋊C2), C4.57(D4⋊2S3), (C2×C12).240C23, (C2×C24).238C22, C12.23D4.4C2, C6.29(C4.4D4), (C2×D12).59C22, C4⋊Dic3.88C22, C2.16(Q8.7D6), C3⋊3(C42.78C22), (C4×Dic3).227C22, C2.19(C23.11D6), (C2×C6).253(C2×D4), (C3×Q8⋊C4)⋊21C2, (C3×C4⋊C4).41C22, (C2×C3⋊C8).217C22, (C2×C4).347(C22×S3), SmallGroup(192,359)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊C4⋊S3
G = < a,b,c,d,e | a4=c4=d3=e2=1, b2=a2, bab-1=cac-1=eae=a-1, ad=da, cbc-1=a-1b, bd=db, ebe=a2bc2, cd=dc, ece=a-1c-1, ede=d-1 >
Subgroups: 296 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×D12, C6×Q8, C42.78C22, C6.D8, C8×Dic3, C2.D24, Q8⋊2Dic3, C3×Q8⋊C4, C4.Dic6, C12.23D4, Q8⋊C4⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C4○D8, C4○D12, S3×D4, D4⋊2S3, C42.78C22, C23.11D6, Q8.7D6, D24⋊C2, Q8⋊C4⋊S3
►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 32 3 30)(2 31 4 29)(5 52 7 50)(6 51 8 49)(9 45 11 47)(10 48 12 46)(13 64 15 62)(14 63 16 61)(17 36 19 34)(18 35 20 33)(21 43 23 41)(22 42 24 44)(25 40 27 38)(26 39 28 37)(53 81 55 83)(54 84 56 82)(57 91 59 89)(58 90 60 92)(65 94 67 96)(66 93 68 95)(69 85 71 87)(70 88 72 86)(73 78 75 80)(74 77 76 79)
(1 54 17 58)(2 53 18 57)(3 56 19 60)(4 55 20 59)(5 73 28 63)(6 76 25 62)(7 75 26 61)(8 74 27 64)(9 71 21 65)(10 70 22 68)(11 69 23 67)(12 72 24 66)(13 52 79 37)(14 51 80 40)(15 50 77 39)(16 49 78 38)(29 84 33 90)(30 83 34 89)(31 82 35 92)(32 81 36 91)(41 93 47 86)(42 96 48 85)(43 95 45 88)(44 94 46 87)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 91 95)(14 92 96)(15 89 93)(16 90 94)(17 21 28)(18 22 25)(19 23 26)(20 24 27)(29 46 49)(30 47 50)(31 48 51)(32 45 52)(33 44 38)(34 41 39)(35 42 40)(36 43 37)(53 70 76)(54 71 73)(55 72 74)(56 69 75)(57 68 62)(58 65 63)(59 66 64)(60 67 61)(77 83 86)(78 84 87)(79 81 88)(80 82 85)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 94)(14 93)(15 96)(16 95)(18 20)(21 28)(22 27)(23 26)(24 25)(29 33)(30 36)(31 35)(32 34)(37 47)(38 46)(39 45)(40 48)(41 52)(42 51)(43 50)(44 49)(53 60)(54 59)(55 58)(56 57)(61 70)(62 69)(63 72)(64 71)(65 74)(66 73)(67 76)(68 75)(77 85)(78 88)(79 87)(80 86)(81 84)(82 83)(89 92)(90 91)
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), (1,32,3,30)(2,31,4,29)(5,52,7,50)(6,51,8,49)(9,45,11,47)(10,48,12,46)(13,64,15,62)(14,63,16,61)(17,36,19,34)(18,35,20,33)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(53,81,55,83)(54,84,56,82)(57,91,59,89)(58,90,60,92)(65,94,67,96)(66,93,68,95)(69,85,71,87)(70,88,72,86)(73,78,75,80)(74,77,76,79), (1,54,17,58)(2,53,18,57)(3,56,19,60)(4,55,20,59)(5,73,28,63)(6,76,25,62)(7,75,26,61)(8,74,27,64)(9,71,21,65)(10,70,22,68)(11,69,23,67)(12,72,24,66)(13,52,79,37)(14,51,80,40)(15,50,77,39)(16,49,78,38)(29,84,33,90)(30,83,34,89)(31,82,35,92)(32,81,36,91)(41,93,47,86)(42,96,48,85)(43,95,45,88)(44,94,46,87), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,91,95)(14,92,96)(15,89,93)(16,90,94)(17,21,28)(18,22,25)(19,23,26)(20,24,27)(29,46,49)(30,47,50)(31,48,51)(32,45,52)(33,44,38)(34,41,39)(35,42,40)(36,43,37)(53,70,76)(54,71,73)(55,72,74)(56,69,75)(57,68,62)(58,65,63)(59,66,64)(60,67,61)(77,83,86)(78,84,87)(79,81,88)(80,82,85), (2,4)(5,9)(6,12)(7,11)(8,10)(13,94)(14,93)(15,96)(16,95)(18,20)(21,28)(22,27)(23,26)(24,25)(29,33)(30,36)(31,35)(32,34)(37,47)(38,46)(39,45)(40,48)(41,52)(42,51)(43,50)(44,49)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,72)(64,71)(65,74)(66,73)(67,76)(68,75)(77,85)(78,88)(79,87)(80,86)(81,84)(82,83)(89,92)(90,91)>;
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), (1,32,3,30)(2,31,4,29)(5,52,7,50)(6,51,8,49)(9,45,11,47)(10,48,12,46)(13,64,15,62)(14,63,16,61)(17,36,19,34)(18,35,20,33)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(53,81,55,83)(54,84,56,82)(57,91,59,89)(58,90,60,92)(65,94,67,96)(66,93,68,95)(69,85,71,87)(70,88,72,86)(73,78,75,80)(74,77,76,79), (1,54,17,58)(2,53,18,57)(3,56,19,60)(4,55,20,59)(5,73,28,63)(6,76,25,62)(7,75,26,61)(8,74,27,64)(9,71,21,65)(10,70,22,68)(11,69,23,67)(12,72,24,66)(13,52,79,37)(14,51,80,40)(15,50,77,39)(16,49,78,38)(29,84,33,90)(30,83,34,89)(31,82,35,92)(32,81,36,91)(41,93,47,86)(42,96,48,85)(43,95,45,88)(44,94,46,87), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,91,95)(14,92,96)(15,89,93)(16,90,94)(17,21,28)(18,22,25)(19,23,26)(20,24,27)(29,46,49)(30,47,50)(31,48,51)(32,45,52)(33,44,38)(34,41,39)(35,42,40)(36,43,37)(53,70,76)(54,71,73)(55,72,74)(56,69,75)(57,68,62)(58,65,63)(59,66,64)(60,67,61)(77,83,86)(78,84,87)(79,81,88)(80,82,85), (2,4)(5,9)(6,12)(7,11)(8,10)(13,94)(14,93)(15,96)(16,95)(18,20)(21,28)(22,27)(23,26)(24,25)(29,33)(30,36)(31,35)(32,34)(37,47)(38,46)(39,45)(40,48)(41,52)(42,51)(43,50)(44,49)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,72)(64,71)(65,74)(66,73)(67,76)(68,75)(77,85)(78,88)(79,87)(80,86)(81,84)(82,83)(89,92)(90,91) );
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)], [(1,32,3,30),(2,31,4,29),(5,52,7,50),(6,51,8,49),(9,45,11,47),(10,48,12,46),(13,64,15,62),(14,63,16,61),(17,36,19,34),(18,35,20,33),(21,43,23,41),(22,42,24,44),(25,40,27,38),(26,39,28,37),(53,81,55,83),(54,84,56,82),(57,91,59,89),(58,90,60,92),(65,94,67,96),(66,93,68,95),(69,85,71,87),(70,88,72,86),(73,78,75,80),(74,77,76,79)], [(1,54,17,58),(2,53,18,57),(3,56,19,60),(4,55,20,59),(5,73,28,63),(6,76,25,62),(7,75,26,61),(8,74,27,64),(9,71,21,65),(10,70,22,68),(11,69,23,67),(12,72,24,66),(13,52,79,37),(14,51,80,40),(15,50,77,39),(16,49,78,38),(29,84,33,90),(30,83,34,89),(31,82,35,92),(32,81,36,91),(41,93,47,86),(42,96,48,85),(43,95,45,88),(44,94,46,87)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,91,95),(14,92,96),(15,89,93),(16,90,94),(17,21,28),(18,22,25),(19,23,26),(20,24,27),(29,46,49),(30,47,50),(31,48,51),(32,45,52),(33,44,38),(34,41,39),(35,42,40),(36,43,37),(53,70,76),(54,71,73),(55,72,74),(56,69,75),(57,68,62),(58,65,63),(59,66,64),(60,67,61),(77,83,86),(78,84,87),(79,81,88),(80,82,85)], [(2,4),(5,9),(6,12),(7,11),(8,10),(13,94),(14,93),(15,96),(16,95),(18,20),(21,28),(22,27),(23,26),(24,25),(29,33),(30,36),(31,35),(32,34),(37,47),(38,46),(39,45),(40,48),(41,52),(42,51),(43,50),(44,49),(53,60),(54,59),(55,58),(56,57),(61,70),(62,69),(63,72),(64,71),(65,74),(66,73),(67,76),(68,75),(77,85),(78,88),(79,87),(80,86),(81,84),(82,83),(89,92),(90,91)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | D4⋊2S3 | S3×D4 | Q8.7D6 | D24⋊C2 |
| kernel | Q8⋊C4⋊S3 | C6.D8 | C8×Dic3 | C2.D24 | Q8⋊2Dic3 | C3×Q8⋊C4 | C4.Dic6 | C12.23D4 | Q8⋊C4 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 8 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Q8⋊C4⋊S3 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 48 | 72 |
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 27 | 8 |
| 0 | 0 | 0 | 46 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 41 | 25 |
| 0 | 0 | 35 | 32 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[30,13,0,0,60,43,0,0,0,0,27,0,0,0,8,46],[66,14,0,0,59,7,0,0,0,0,41,35,0,0,25,32],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,48,0,0,0,72] >;
Q8⋊C4⋊S3 in GAP, Magma, Sage, TeX
Q_8\rtimes C_4\rtimes S_3
% in TeX
G:=Group("Q8:C4:S3"); // GroupNames label
G:=SmallGroup(192,359);
// by ID
G=gap.SmallGroup(192,359);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,120,1094,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^4=d^3=e^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,e*b*e=a^2*b*c^2,c*d=d*c,e*c*e=a^-1*c^-1,e*d*e=d^-1>;
// generators/relations