metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.63D6, (C2×C12).75D4, (C2×Q8).49D6, C22⋊Q8.2S3, (C2×C6).16SD16, C6.71(C2×SD16), (C22×C6).89D4, Q8⋊2Dic3⋊12C2, C12.Q8⋊37C2, (C22×C4).140D6, (C6×Q8).43C22, C12.187(C4○D4), C4.93(D4⋊2S3), (C2×C12).362C23, C12.55D4.6C2, C23.67(C3⋊D4), C3⋊5(C23.47D4), C2.14(Q8.14D6), C6.116(C8.C22), C4⋊Dic3.337C22, C22.6(Q8⋊2S3), (C22×C12).166C22, C6.80(C22.D4), C2.14(C23.23D6), (C2×C6).493(C2×D4), C2.8(C2×Q8⋊2S3), (C3×C22⋊Q8).1C2, (C2×C4).53(C3⋊D4), (C2×C3⋊C8).112C22, (C2×C4⋊Dic3).37C2, (C3×C4⋊C4).110C22, (C2×C4).462(C22×S3), C22.168(C2×C3⋊D4), SmallGroup(192,602)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Q8).49D6
G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=ab2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 256 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.47D4, C12.Q8, C12.55D4, Q8⋊2Dic3, C2×C4⋊Dic3, C3×C22⋊Q8, (C2×Q8).49D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×SD16, C8.C22, Q8⋊2S3, D4⋊2S3, C2×C3⋊D4, C23.47D4, C23.23D6, C2×Q8⋊2S3, Q8.14D6, (C2×Q8).49D6
►On 96 points
Generators in S96
(1 10)(2 11)(3 12)(4 25)(5 26)(6 27)(7 44)(8 45)(9 43)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(28 36)(29 34)(30 35)(31 39)(32 37)(33 38)(40 48)(41 46)(42 47)(49 86)(50 87)(51 88)(52 89)(53 90)(54 85)(55 65)(56 66)(57 61)(58 62)(59 63)(60 64)(67 84)(68 79)(69 80)(70 81)(71 82)(72 83)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)
(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 48 28 32)(8 46 29 33)(9 47 30 31)(10 13 25 16)(11 14 26 17)(12 15 27 18)(34 38 45 41)(35 39 43 42)(36 37 44 40)(49 64 89 57)(50 65 90 58)(51 66 85 59)(52 61 86 60)(53 62 87 55)(54 63 88 56)(67 77 81 91)(68 78 82 92)(69 73 83 93)(70 74 84 94)(71 75 79 95)(72 76 80 96)
(1 52 4 86)(2 50 5 90)(3 54 6 88)(7 77 28 91)(8 75 29 95)(9 73 30 93)(10 89 25 49)(11 87 26 53)(12 85 27 51)(13 64 16 57)(14 62 17 55)(15 66 18 59)(19 61 22 60)(20 65 23 58)(21 63 24 56)(31 83 47 69)(32 81 48 67)(33 79 46 71)(34 78 45 92)(35 76 43 96)(36 74 44 94)(37 70 40 84)(38 68 41 82)(39 72 42 80)
(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 25 8)(2 36 26 7)(3 35 27 9)(4 45 10 29)(5 44 11 28)(6 43 12 30)(13 46 19 38)(14 48 20 37)(15 47 21 39)(16 33 22 41)(17 32 23 40)(18 31 24 42)(49 79 52 82)(50 84 53 81)(51 83 54 80)(55 91 58 94)(56 96 59 93)(57 95 60 92)(61 78 64 75)(62 77 65 74)(63 76 66 73)(67 90 70 87)(68 89 71 86)(69 88 72 85)
G:=sub<Sym(96)| (1,10)(2,11)(3,12)(4,25)(5,26)(6,27)(7,44)(8,45)(9,43)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(28,36)(29,34)(30,35)(31,39)(32,37)(33,38)(40,48)(41,46)(42,47)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(55,65)(56,66)(57,61)(58,62)(59,63)(60,64)(67,84)(68,79)(69,80)(70,81)(71,82)(72,83)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,48,28,32)(8,46,29,33)(9,47,30,31)(10,13,25,16)(11,14,26,17)(12,15,27,18)(34,38,45,41)(35,39,43,42)(36,37,44,40)(49,64,89,57)(50,65,90,58)(51,66,85,59)(52,61,86,60)(53,62,87,55)(54,63,88,56)(67,77,81,91)(68,78,82,92)(69,73,83,93)(70,74,84,94)(71,75,79,95)(72,76,80,96), (1,52,4,86)(2,50,5,90)(3,54,6,88)(7,77,28,91)(8,75,29,95)(9,73,30,93)(10,89,25,49)(11,87,26,53)(12,85,27,51)(13,64,16,57)(14,62,17,55)(15,66,18,59)(19,61,22,60)(20,65,23,58)(21,63,24,56)(31,83,47,69)(32,81,48,67)(33,79,46,71)(34,78,45,92)(35,76,43,96)(36,74,44,94)(37,70,40,84)(38,68,41,82)(39,72,42,80), (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), (1,34,25,8)(2,36,26,7)(3,35,27,9)(4,45,10,29)(5,44,11,28)(6,43,12,30)(13,46,19,38)(14,48,20,37)(15,47,21,39)(16,33,22,41)(17,32,23,40)(18,31,24,42)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,91,58,94)(56,96,59,93)(57,95,60,92)(61,78,64,75)(62,77,65,74)(63,76,66,73)(67,90,70,87)(68,89,71,86)(69,88,72,85)>;
G:=Group( (1,10)(2,11)(3,12)(4,25)(5,26)(6,27)(7,44)(8,45)(9,43)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(28,36)(29,34)(30,35)(31,39)(32,37)(33,38)(40,48)(41,46)(42,47)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(55,65)(56,66)(57,61)(58,62)(59,63)(60,64)(67,84)(68,79)(69,80)(70,81)(71,82)(72,83)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,48,28,32)(8,46,29,33)(9,47,30,31)(10,13,25,16)(11,14,26,17)(12,15,27,18)(34,38,45,41)(35,39,43,42)(36,37,44,40)(49,64,89,57)(50,65,90,58)(51,66,85,59)(52,61,86,60)(53,62,87,55)(54,63,88,56)(67,77,81,91)(68,78,82,92)(69,73,83,93)(70,74,84,94)(71,75,79,95)(72,76,80,96), (1,52,4,86)(2,50,5,90)(3,54,6,88)(7,77,28,91)(8,75,29,95)(9,73,30,93)(10,89,25,49)(11,87,26,53)(12,85,27,51)(13,64,16,57)(14,62,17,55)(15,66,18,59)(19,61,22,60)(20,65,23,58)(21,63,24,56)(31,83,47,69)(32,81,48,67)(33,79,46,71)(34,78,45,92)(35,76,43,96)(36,74,44,94)(37,70,40,84)(38,68,41,82)(39,72,42,80), (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), (1,34,25,8)(2,36,26,7)(3,35,27,9)(4,45,10,29)(5,44,11,28)(6,43,12,30)(13,46,19,38)(14,48,20,37)(15,47,21,39)(16,33,22,41)(17,32,23,40)(18,31,24,42)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,91,58,94)(56,96,59,93)(57,95,60,92)(61,78,64,75)(62,77,65,74)(63,76,66,73)(67,90,70,87)(68,89,71,86)(69,88,72,85) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,25),(5,26),(6,27),(7,44),(8,45),(9,43),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(28,36),(29,34),(30,35),(31,39),(32,37),(33,38),(40,48),(41,46),(42,47),(49,86),(50,87),(51,88),(52,89),(53,90),(54,85),(55,65),(56,66),(57,61),(58,62),(59,63),(60,64),(67,84),(68,79),(69,80),(70,81),(71,82),(72,83),(73,96),(74,91),(75,92),(76,93),(77,94),(78,95)], [(1,22,4,19),(2,23,5,20),(3,24,6,21),(7,48,28,32),(8,46,29,33),(9,47,30,31),(10,13,25,16),(11,14,26,17),(12,15,27,18),(34,38,45,41),(35,39,43,42),(36,37,44,40),(49,64,89,57),(50,65,90,58),(51,66,85,59),(52,61,86,60),(53,62,87,55),(54,63,88,56),(67,77,81,91),(68,78,82,92),(69,73,83,93),(70,74,84,94),(71,75,79,95),(72,76,80,96)], [(1,52,4,86),(2,50,5,90),(3,54,6,88),(7,77,28,91),(8,75,29,95),(9,73,30,93),(10,89,25,49),(11,87,26,53),(12,85,27,51),(13,64,16,57),(14,62,17,55),(15,66,18,59),(19,61,22,60),(20,65,23,58),(21,63,24,56),(31,83,47,69),(32,81,48,67),(33,79,46,71),(34,78,45,92),(35,76,43,96),(36,74,44,94),(37,70,40,84),(38,68,41,82),(39,72,42,80)], [(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)], [(1,34,25,8),(2,36,26,7),(3,35,27,9),(4,45,10,29),(5,44,11,28),(6,43,12,30),(13,46,19,38),(14,48,20,37),(15,47,21,39),(16,33,22,41),(17,32,23,40),(18,31,24,42),(49,79,52,82),(50,84,53,81),(51,83,54,80),(55,91,58,94),(56,96,59,93),(57,95,60,92),(61,78,64,75),(62,77,65,74),(63,76,66,73),(67,90,70,87),(68,89,71,86),(69,88,72,85)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | SD16 | C3⋊D4 | C3⋊D4 | C8.C22 | D4⋊2S3 | Q8⋊2S3 | Q8.14D6 |
| kernel | (C2×Q8).49D6 | C12.Q8 | C12.55D4 | Q8⋊2Dic3 | C2×C4⋊Dic3 | C3×C22⋊Q8 | C22⋊Q8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of (C2×Q8).49D6 ►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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 3 |
| 0 | 0 | 0 | 0 | 23 | 52 |
| 72 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 16 |
| 0 | 0 | 0 | 0 | 39 | 55 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 5 | 0 | 0 |
| 0 | 0 | 23 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 37 |
| 0 | 0 | 0 | 0 | 6 | 12 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,21,23,0,0,0,0,3,52],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,18,39,0,0,0,0,16,55],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,27,0,0,0,0,0,46,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,61,6,0,0,0,0,37,12] >;
(C2×Q8).49D6 in GAP, Magma, Sage, TeX
(C_2\times Q_8)._{49}D_6 % in TeX
G:=Group("(C2xQ8).49D6"); // GroupNames label
G:=SmallGroup(192,602);
// by ID
G=gap.SmallGroup(192,602);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=a*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^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations