direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C42⋊7S3, C42⋊42D6, (C2×C42)⋊11S3, C4.43(C2×D12), (C2×C4).98D12, C6.4(C22×D4), (C4×C12)⋊51C22, D6⋊C4⋊39C22, (C2×C12).389D4, C12.286(C2×D4), C6⋊1(C4.4D4), (C2×C6).20C24, C2.6(C22×D12), (C22×Dic6)⋊4C2, (C22×D12).7C2, C22.65(C2×D12), (C22×C4).455D6, (C2×C12).781C23, (C2×Dic6)⋊46C22, (C22×S3).2C23, C22.63(S3×C23), (C2×Dic3).4C23, (C2×D12).202C22, C22.69(C4○D12), (S3×C23).28C22, C23.326(C22×S3), (C22×C6).382C23, (C22×C12).504C22, (C22×Dic3).74C22, (C2×C4×C12)⋊9C2, C6.7(C2×C4○D4), C3⋊1(C2×C4.4D4), (C2×D6⋊C4)⋊12C2, C2.9(C2×C4○D12), (C2×C6).171(C2×D4), (C2×C6).97(C4○D4), (C2×C4).649(C22×S3), SmallGroup(192,1035)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C42⋊7S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 952 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, D6⋊C4, C4×C12, C2×Dic6, C2×Dic6, C2×D12, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, C2×C4.4D4, C42⋊7S3, C2×D6⋊C4, C2×C4×C12, C22×Dic6, C22×D12, C2×C42⋊7S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C4.4D4, C22×D4, C2×C4○D4, C2×D12, C4○D12, S3×C23, C2×C4.4D4, C42⋊7S3, C22×D12, C2×C4○D12, C2×C42⋊7S3
►On 96 points
Generators in S96
(1 64)(2 61)(3 62)(4 63)(5 85)(6 86)(7 87)(8 88)(9 24)(10 21)(11 22)(12 23)(13 75)(14 76)(15 73)(16 74)(17 77)(18 78)(19 79)(20 80)(25 44)(26 41)(27 42)(28 43)(29 68)(30 65)(31 66)(32 67)(33 38)(34 39)(35 40)(36 37)(45 52)(46 49)(47 50)(48 51)(53 59)(54 60)(55 57)(56 58)(69 90)(70 91)(71 92)(72 89)(81 96)(82 93)(83 94)(84 95)
(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 74 91 29)(2 75 92 30)(3 76 89 31)(4 73 90 32)(5 22 28 34)(6 23 25 35)(7 24 26 36)(8 21 27 33)(9 41 37 87)(10 42 38 88)(11 43 39 85)(12 44 40 86)(13 71 65 61)(14 72 66 62)(15 69 67 63)(16 70 68 64)(17 83 60 47)(18 84 57 48)(19 81 58 45)(20 82 59 46)(49 80 93 53)(50 77 94 54)(51 78 95 55)(52 79 96 56)
(1 51 23)(2 52 24)(3 49 21)(4 50 22)(5 32 54)(6 29 55)(7 30 56)(8 31 53)(9 61 45)(10 62 46)(11 63 47)(12 64 48)(13 19 41)(14 20 42)(15 17 43)(16 18 44)(25 74 78)(26 75 79)(27 76 80)(28 73 77)(33 89 93)(34 90 94)(35 91 95)(36 92 96)(37 71 81)(38 72 82)(39 69 83)(40 70 84)(57 86 68)(58 87 65)(59 88 66)(60 85 67)
(1 65)(2 14)(3 67)(4 16)(5 46)(6 83)(7 48)(8 81)(9 80)(10 54)(11 78)(12 56)(13 91)(15 89)(17 33)(18 22)(19 35)(20 24)(21 60)(23 58)(25 47)(26 84)(27 45)(28 82)(29 69)(30 64)(31 71)(32 62)(34 57)(36 59)(37 53)(38 77)(39 55)(40 79)(41 95)(42 52)(43 93)(44 50)(49 85)(51 87)(61 76)(63 74)(66 92)(68 90)(70 75)(72 73)(86 94)(88 96)
G:=sub<Sym(96)| (1,64)(2,61)(3,62)(4,63)(5,85)(6,86)(7,87)(8,88)(9,24)(10,21)(11,22)(12,23)(13,75)(14,76)(15,73)(16,74)(17,77)(18,78)(19,79)(20,80)(25,44)(26,41)(27,42)(28,43)(29,68)(30,65)(31,66)(32,67)(33,38)(34,39)(35,40)(36,37)(45,52)(46,49)(47,50)(48,51)(53,59)(54,60)(55,57)(56,58)(69,90)(70,91)(71,92)(72,89)(81,96)(82,93)(83,94)(84,95), (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), (1,74,91,29)(2,75,92,30)(3,76,89,31)(4,73,90,32)(5,22,28,34)(6,23,25,35)(7,24,26,36)(8,21,27,33)(9,41,37,87)(10,42,38,88)(11,43,39,85)(12,44,40,86)(13,71,65,61)(14,72,66,62)(15,69,67,63)(16,70,68,64)(17,83,60,47)(18,84,57,48)(19,81,58,45)(20,82,59,46)(49,80,93,53)(50,77,94,54)(51,78,95,55)(52,79,96,56), (1,51,23)(2,52,24)(3,49,21)(4,50,22)(5,32,54)(6,29,55)(7,30,56)(8,31,53)(9,61,45)(10,62,46)(11,63,47)(12,64,48)(13,19,41)(14,20,42)(15,17,43)(16,18,44)(25,74,78)(26,75,79)(27,76,80)(28,73,77)(33,89,93)(34,90,94)(35,91,95)(36,92,96)(37,71,81)(38,72,82)(39,69,83)(40,70,84)(57,86,68)(58,87,65)(59,88,66)(60,85,67), (1,65)(2,14)(3,67)(4,16)(5,46)(6,83)(7,48)(8,81)(9,80)(10,54)(11,78)(12,56)(13,91)(15,89)(17,33)(18,22)(19,35)(20,24)(21,60)(23,58)(25,47)(26,84)(27,45)(28,82)(29,69)(30,64)(31,71)(32,62)(34,57)(36,59)(37,53)(38,77)(39,55)(40,79)(41,95)(42,52)(43,93)(44,50)(49,85)(51,87)(61,76)(63,74)(66,92)(68,90)(70,75)(72,73)(86,94)(88,96)>;
G:=Group( (1,64)(2,61)(3,62)(4,63)(5,85)(6,86)(7,87)(8,88)(9,24)(10,21)(11,22)(12,23)(13,75)(14,76)(15,73)(16,74)(17,77)(18,78)(19,79)(20,80)(25,44)(26,41)(27,42)(28,43)(29,68)(30,65)(31,66)(32,67)(33,38)(34,39)(35,40)(36,37)(45,52)(46,49)(47,50)(48,51)(53,59)(54,60)(55,57)(56,58)(69,90)(70,91)(71,92)(72,89)(81,96)(82,93)(83,94)(84,95), (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), (1,74,91,29)(2,75,92,30)(3,76,89,31)(4,73,90,32)(5,22,28,34)(6,23,25,35)(7,24,26,36)(8,21,27,33)(9,41,37,87)(10,42,38,88)(11,43,39,85)(12,44,40,86)(13,71,65,61)(14,72,66,62)(15,69,67,63)(16,70,68,64)(17,83,60,47)(18,84,57,48)(19,81,58,45)(20,82,59,46)(49,80,93,53)(50,77,94,54)(51,78,95,55)(52,79,96,56), (1,51,23)(2,52,24)(3,49,21)(4,50,22)(5,32,54)(6,29,55)(7,30,56)(8,31,53)(9,61,45)(10,62,46)(11,63,47)(12,64,48)(13,19,41)(14,20,42)(15,17,43)(16,18,44)(25,74,78)(26,75,79)(27,76,80)(28,73,77)(33,89,93)(34,90,94)(35,91,95)(36,92,96)(37,71,81)(38,72,82)(39,69,83)(40,70,84)(57,86,68)(58,87,65)(59,88,66)(60,85,67), (1,65)(2,14)(3,67)(4,16)(5,46)(6,83)(7,48)(8,81)(9,80)(10,54)(11,78)(12,56)(13,91)(15,89)(17,33)(18,22)(19,35)(20,24)(21,60)(23,58)(25,47)(26,84)(27,45)(28,82)(29,69)(30,64)(31,71)(32,62)(34,57)(36,59)(37,53)(38,77)(39,55)(40,79)(41,95)(42,52)(43,93)(44,50)(49,85)(51,87)(61,76)(63,74)(66,92)(68,90)(70,75)(72,73)(86,94)(88,96) );
G=PermutationGroup([[(1,64),(2,61),(3,62),(4,63),(5,85),(6,86),(7,87),(8,88),(9,24),(10,21),(11,22),(12,23),(13,75),(14,76),(15,73),(16,74),(17,77),(18,78),(19,79),(20,80),(25,44),(26,41),(27,42),(28,43),(29,68),(30,65),(31,66),(32,67),(33,38),(34,39),(35,40),(36,37),(45,52),(46,49),(47,50),(48,51),(53,59),(54,60),(55,57),(56,58),(69,90),(70,91),(71,92),(72,89),(81,96),(82,93),(83,94),(84,95)], [(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)], [(1,74,91,29),(2,75,92,30),(3,76,89,31),(4,73,90,32),(5,22,28,34),(6,23,25,35),(7,24,26,36),(8,21,27,33),(9,41,37,87),(10,42,38,88),(11,43,39,85),(12,44,40,86),(13,71,65,61),(14,72,66,62),(15,69,67,63),(16,70,68,64),(17,83,60,47),(18,84,57,48),(19,81,58,45),(20,82,59,46),(49,80,93,53),(50,77,94,54),(51,78,95,55),(52,79,96,56)], [(1,51,23),(2,52,24),(3,49,21),(4,50,22),(5,32,54),(6,29,55),(7,30,56),(8,31,53),(9,61,45),(10,62,46),(11,63,47),(12,64,48),(13,19,41),(14,20,42),(15,17,43),(16,18,44),(25,74,78),(26,75,79),(27,76,80),(28,73,77),(33,89,93),(34,90,94),(35,91,95),(36,92,96),(37,71,81),(38,72,82),(39,69,83),(40,70,84),(57,86,68),(58,87,65),(59,88,66),(60,85,67)], [(1,65),(2,14),(3,67),(4,16),(5,46),(6,83),(7,48),(8,81),(9,80),(10,54),(11,78),(12,56),(13,91),(15,89),(17,33),(18,22),(19,35),(20,24),(21,60),(23,58),(25,47),(26,84),(27,45),(28,82),(29,69),(30,64),(31,71),(32,62),(34,57),(36,59),(37,53),(38,77),(39,55),(40,79),(41,95),(42,52),(43,93),(44,50),(49,85),(51,87),(61,76),(63,74),(66,92),(68,90),(70,75),(72,73),(86,94),(88,96)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C4○D12 |
| kernel | C2×C42⋊7S3 | C42⋊7S3 | C2×D6⋊C4 | C2×C4×C12 | C22×Dic6 | C22×D12 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 8 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 16 |
Matrix representation of C2×C42⋊7S3 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 7 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[8,5,0,0,0,0,3,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;
C2×C42⋊7S3 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_7S_3
% in TeX
G:=Group("C2xC4^2:7S3"); // GroupNames label
G:=SmallGroup(192,1035);
// by ID
G=gap.SmallGroup(192,1035);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations