metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Q8⋊9S3, D6⋊C8⋊31C2, C4⋊C4.40D6, (C2×C8).139D6, C12⋊D4.6C2, C6.56(C4○D8), C2.D24⋊32C2, C6.Q16⋊16C2, C6.D8⋊17C2, C12.31(C4○D4), C4.74(C4○D12), C2.23(Q8⋊3D6), C6.71(C8⋊C22), (C22×S3).25D4, C22.218(S3×D4), (C2×C24).286C22, (C2×C12).282C23, C4.26(Q8⋊3S3), (C2×Dic3).163D4, (C2×D12).76C22, C3⋊4(C23.19D4), C2.23(Q8.7D6), C2.13(D6.D4), C4⋊Dic3.112C22, C6.43(C22.D4), C4⋊C4⋊7S3⋊6C2, (C3×C4.Q8)⋊17C2, (C2×C6).287(C2×D4), (C2×C3⋊C8).59C22, (S3×C2×C4).34C22, (C3×C4⋊C4).75C22, (C2×C4).385(C22×S3), SmallGroup(192,425)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Q8⋊S3
G = < a,b,c,d,e | a4=d3=e2=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, ebe=a-1b3, cd=dc, ece=a2c, ede=d-1 >
Subgroups: 368 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C23.19D4, C6.Q16, C6.D8, D6⋊C8, C2.D24, C3×C4.Q8, C4⋊C4⋊7S3, C12⋊D4, C4.Q8⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8⋊C22, C4○D12, S3×D4, Q8⋊3S3, C23.19D4, D6.D4, Q8⋊3D6, Q8.7D6, C4.Q8⋊S3
►On 96 points
Generators in S96
(1 79 5 75)(2 80 6 76)(3 73 7 77)(4 74 8 78)(9 95 13 91)(10 96 14 92)(11 89 15 93)(12 90 16 94)(17 62 21 58)(18 63 22 59)(19 64 23 60)(20 57 24 61)(25 67 29 71)(26 68 30 72)(27 69 31 65)(28 70 32 66)(33 86 37 82)(34 87 38 83)(35 88 39 84)(36 81 40 85)(41 55 45 51)(42 56 46 52)(43 49 47 53)(44 50 48 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 9 77 93)(2 12 78 96)(3 15 79 91)(4 10 80 94)(5 13 73 89)(6 16 74 92)(7 11 75 95)(8 14 76 90)(17 33 60 84)(18 36 61 87)(19 39 62 82)(20 34 63 85)(21 37 64 88)(22 40 57 83)(23 35 58 86)(24 38 59 81)(25 50 65 46)(26 53 66 41)(27 56 67 44)(28 51 68 47)(29 54 69 42)(30 49 70 45)(31 52 71 48)(32 55 72 43)
(1 53 19)(2 54 20)(3 55 21)(4 56 22)(5 49 23)(6 50 24)(7 51 17)(8 52 18)(9 66 39)(10 67 40)(11 68 33)(12 69 34)(13 70 35)(14 71 36)(15 72 37)(16 65 38)(25 81 92)(26 82 93)(27 83 94)(28 84 95)(29 85 96)(30 86 89)(31 87 90)(32 88 91)(41 62 77)(42 63 78)(43 64 79)(44 57 80)(45 58 73)(46 59 74)(47 60 75)(48 61 76)
(1 90)(2 15)(3 92)(4 9)(5 94)(6 11)(7 96)(8 13)(10 73)(12 75)(14 77)(16 79)(17 29)(18 70)(19 31)(20 72)(21 25)(22 66)(23 27)(24 68)(26 57)(28 59)(30 61)(32 63)(33 50)(34 47)(35 52)(36 41)(37 54)(38 43)(39 56)(40 45)(42 88)(44 82)(46 84)(48 86)(49 83)(51 85)(53 87)(55 81)(58 67)(60 69)(62 71)(64 65)(74 95)(76 89)(78 91)(80 93)
G:=sub<Sym(96)| (1,79,5,75)(2,80,6,76)(3,73,7,77)(4,74,8,78)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,62,21,58)(18,63,22,59)(19,64,23,60)(20,57,24,61)(25,67,29,71)(26,68,30,72)(27,69,31,65)(28,70,32,66)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(41,55,45,51)(42,56,46,52)(43,49,47,53)(44,50,48,54), (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), (1,9,77,93)(2,12,78,96)(3,15,79,91)(4,10,80,94)(5,13,73,89)(6,16,74,92)(7,11,75,95)(8,14,76,90)(17,33,60,84)(18,36,61,87)(19,39,62,82)(20,34,63,85)(21,37,64,88)(22,40,57,83)(23,35,58,86)(24,38,59,81)(25,50,65,46)(26,53,66,41)(27,56,67,44)(28,51,68,47)(29,54,69,42)(30,49,70,45)(31,52,71,48)(32,55,72,43), (1,53,19)(2,54,20)(3,55,21)(4,56,22)(5,49,23)(6,50,24)(7,51,17)(8,52,18)(9,66,39)(10,67,40)(11,68,33)(12,69,34)(13,70,35)(14,71,36)(15,72,37)(16,65,38)(25,81,92)(26,82,93)(27,83,94)(28,84,95)(29,85,96)(30,86,89)(31,87,90)(32,88,91)(41,62,77)(42,63,78)(43,64,79)(44,57,80)(45,58,73)(46,59,74)(47,60,75)(48,61,76), (1,90)(2,15)(3,92)(4,9)(5,94)(6,11)(7,96)(8,13)(10,73)(12,75)(14,77)(16,79)(17,29)(18,70)(19,31)(20,72)(21,25)(22,66)(23,27)(24,68)(26,57)(28,59)(30,61)(32,63)(33,50)(34,47)(35,52)(36,41)(37,54)(38,43)(39,56)(40,45)(42,88)(44,82)(46,84)(48,86)(49,83)(51,85)(53,87)(55,81)(58,67)(60,69)(62,71)(64,65)(74,95)(76,89)(78,91)(80,93)>;
G:=Group( (1,79,5,75)(2,80,6,76)(3,73,7,77)(4,74,8,78)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,62,21,58)(18,63,22,59)(19,64,23,60)(20,57,24,61)(25,67,29,71)(26,68,30,72)(27,69,31,65)(28,70,32,66)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(41,55,45,51)(42,56,46,52)(43,49,47,53)(44,50,48,54), (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), (1,9,77,93)(2,12,78,96)(3,15,79,91)(4,10,80,94)(5,13,73,89)(6,16,74,92)(7,11,75,95)(8,14,76,90)(17,33,60,84)(18,36,61,87)(19,39,62,82)(20,34,63,85)(21,37,64,88)(22,40,57,83)(23,35,58,86)(24,38,59,81)(25,50,65,46)(26,53,66,41)(27,56,67,44)(28,51,68,47)(29,54,69,42)(30,49,70,45)(31,52,71,48)(32,55,72,43), (1,53,19)(2,54,20)(3,55,21)(4,56,22)(5,49,23)(6,50,24)(7,51,17)(8,52,18)(9,66,39)(10,67,40)(11,68,33)(12,69,34)(13,70,35)(14,71,36)(15,72,37)(16,65,38)(25,81,92)(26,82,93)(27,83,94)(28,84,95)(29,85,96)(30,86,89)(31,87,90)(32,88,91)(41,62,77)(42,63,78)(43,64,79)(44,57,80)(45,58,73)(46,59,74)(47,60,75)(48,61,76), (1,90)(2,15)(3,92)(4,9)(5,94)(6,11)(7,96)(8,13)(10,73)(12,75)(14,77)(16,79)(17,29)(18,70)(19,31)(20,72)(21,25)(22,66)(23,27)(24,68)(26,57)(28,59)(30,61)(32,63)(33,50)(34,47)(35,52)(36,41)(37,54)(38,43)(39,56)(40,45)(42,88)(44,82)(46,84)(48,86)(49,83)(51,85)(53,87)(55,81)(58,67)(60,69)(62,71)(64,65)(74,95)(76,89)(78,91)(80,93) );
G=PermutationGroup([[(1,79,5,75),(2,80,6,76),(3,73,7,77),(4,74,8,78),(9,95,13,91),(10,96,14,92),(11,89,15,93),(12,90,16,94),(17,62,21,58),(18,63,22,59),(19,64,23,60),(20,57,24,61),(25,67,29,71),(26,68,30,72),(27,69,31,65),(28,70,32,66),(33,86,37,82),(34,87,38,83),(35,88,39,84),(36,81,40,85),(41,55,45,51),(42,56,46,52),(43,49,47,53),(44,50,48,54)], [(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)], [(1,9,77,93),(2,12,78,96),(3,15,79,91),(4,10,80,94),(5,13,73,89),(6,16,74,92),(7,11,75,95),(8,14,76,90),(17,33,60,84),(18,36,61,87),(19,39,62,82),(20,34,63,85),(21,37,64,88),(22,40,57,83),(23,35,58,86),(24,38,59,81),(25,50,65,46),(26,53,66,41),(27,56,67,44),(28,51,68,47),(29,54,69,42),(30,49,70,45),(31,52,71,48),(32,55,72,43)], [(1,53,19),(2,54,20),(3,55,21),(4,56,22),(5,49,23),(6,50,24),(7,51,17),(8,52,18),(9,66,39),(10,67,40),(11,68,33),(12,69,34),(13,70,35),(14,71,36),(15,72,37),(16,65,38),(25,81,92),(26,82,93),(27,83,94),(28,84,95),(29,85,96),(30,86,89),(31,87,90),(32,88,91),(41,62,77),(42,63,78),(43,64,79),(44,57,80),(45,58,73),(46,59,74),(47,60,75),(48,61,76)], [(1,90),(2,15),(3,92),(4,9),(5,94),(6,11),(7,96),(8,13),(10,73),(12,75),(14,77),(16,79),(17,29),(18,70),(19,31),(20,72),(21,25),(22,66),(23,27),(24,68),(26,57),(28,59),(30,61),(32,63),(33,50),(34,47),(35,52),(36,41),(37,54),(38,43),(39,56),(40,45),(42,88),(44,82),(46,84),(48,86),(49,83),(51,85),(53,87),(55,81),(58,67),(60,69),(62,71),(64,65),(74,95),(76,89),(78,91),(80,93)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 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 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8⋊C22 | Q8⋊3S3 | S3×D4 | Q8⋊3D6 | Q8.7D6 |
| kernel | C4.Q8⋊S3 | C6.Q16 | C6.D8 | D6⋊C8 | C2.D24 | C3×C4.Q8 | C4⋊C4⋊7S3 | C12⋊D4 | C4.Q8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C4.Q8⋊S3 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 46 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,6,6,0,0,0,0,67,6],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,57,0,0,0,0,57,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0] >;
C4.Q8⋊S3 in GAP, Magma, Sage, TeX
C_4.Q_8\rtimes S_3
% in TeX
G:=Group("C4.Q8:S3"); // GroupNames label
G:=SmallGroup(192,425);
// by ID
G=gap.SmallGroup(192,425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,254,219,100,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^3=e^2=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,e*b*e=a^-1*b^3,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations