metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊1C4, (C2×C42)⋊3S3, C6.63(C4×D4), (C2×C4).66D12, C2.20(C4×D12), (C2×C12).448D4, C6.57(C4⋊D4), C2.3(C12⋊7D4), C6.C42⋊8C2, (C22×C4).416D6, C22.40(C2×D12), C6.5(C42⋊2C2), C2.4(C42⋊7S3), C2.3(C42⋊3S3), C6.13(C4.4D4), C6.15(C42⋊C2), C2.15(C42⋊2S3), C22.49(C4○D12), (S3×C23).12C22, (C22×C6).315C23, C23.283(C22×S3), C3⋊3(C24.C22), (C22×C12).479C22, C6.56(C22.D4), C2.2(C23.28D6), (C22×Dic3).33C22, (C2×C4×C12)⋊1C2, C2.6(C4×C3⋊D4), (C2×C4).92(C4×S3), (C2×Dic3⋊C4)⋊5C2, (C2×D6⋊C4).10C2, (C2×C6).148(C2×D4), C22.120(S3×C2×C4), (C2×C12).208(C2×C4), (C2×C6).74(C4○D4), C22.44(C2×C3⋊D4), (C2×C4).213(C3⋊D4), (C22×S3).19(C2×C4), (C2×C6).101(C22×C4), (C2×Dic3).27(C2×C4), SmallGroup(192,499)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C42)⋊3S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ebe=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 536 in 190 conjugacy classes, 71 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C4×C12, C22×Dic3, C22×C12, S3×C23, C24.C22, C6.C42, C2×Dic3⋊C4, C2×D6⋊C4, C2×C4×C12, (C2×C42)⋊3S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C24.C22, C42⋊2S3, C4×D12, C42⋊7S3, C42⋊3S3, C4×C3⋊D4, C23.28D6, C12⋊7D4, (C2×C42)⋊3S3
►On 96 points
Generators in S96
(1 53)(2 54)(3 55)(4 56)(5 79)(6 80)(7 77)(8 78)(9 29)(10 30)(11 31)(12 32)(13 35)(14 36)(15 33)(16 34)(17 91)(18 92)(19 89)(20 90)(21 41)(22 42)(23 43)(24 44)(25 47)(26 48)(27 45)(28 46)(37 59)(38 60)(39 57)(40 58)(49 71)(50 72)(51 69)(52 70)(61 81)(62 82)(63 83)(64 84)(65 87)(66 88)(67 85)(68 86)(73 93)(74 94)(75 95)(76 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 47 35 87)(2 48 36 88)(3 45 33 85)(4 46 34 86)(5 69 57 17)(6 70 58 18)(7 71 59 19)(8 72 60 20)(9 73 61 21)(10 74 62 22)(11 75 63 23)(12 76 64 24)(13 65 53 25)(14 66 54 26)(15 67 55 27)(16 68 56 28)(29 93 81 41)(30 94 82 42)(31 95 83 43)(32 96 84 44)(37 89 77 49)(38 90 78 50)(39 91 79 51)(40 92 80 52)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 81 39)(14 82 40)(15 83 37)(16 84 38)(17 87 21)(18 88 22)(19 85 23)(20 86 24)(25 93 51)(26 94 52)(27 95 49)(28 96 50)(29 79 53)(30 80 54)(31 77 55)(32 78 56)(33 63 59)(34 64 60)(35 61 57)(36 62 58)(41 91 65)(42 92 66)(43 89 67)(44 90 68)(45 75 71)(46 76 72)(47 73 69)(48 74 70)
(2 54)(4 56)(5 9)(6 30)(7 11)(8 32)(10 80)(12 78)(14 36)(16 34)(17 23)(18 44)(19 21)(20 42)(22 90)(24 92)(25 27)(26 46)(28 48)(29 79)(31 77)(37 83)(38 64)(39 81)(40 62)(41 89)(43 91)(45 47)(49 93)(50 74)(51 95)(52 76)(57 61)(58 82)(59 63)(60 84)(65 67)(66 86)(68 88)(69 75)(70 96)(71 73)(72 94)(85 87)
G:=sub<Sym(96)| (1,53)(2,54)(3,55)(4,56)(5,79)(6,80)(7,77)(8,78)(9,29)(10,30)(11,31)(12,32)(13,35)(14,36)(15,33)(16,34)(17,91)(18,92)(19,89)(20,90)(21,41)(22,42)(23,43)(24,44)(25,47)(26,48)(27,45)(28,46)(37,59)(38,60)(39,57)(40,58)(49,71)(50,72)(51,69)(52,70)(61,81)(62,82)(63,83)(64,84)(65,87)(66,88)(67,85)(68,86)(73,93)(74,94)(75,95)(76,96), (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), (1,47,35,87)(2,48,36,88)(3,45,33,85)(4,46,34,86)(5,69,57,17)(6,70,58,18)(7,71,59,19)(8,72,60,20)(9,73,61,21)(10,74,62,22)(11,75,63,23)(12,76,64,24)(13,65,53,25)(14,66,54,26)(15,67,55,27)(16,68,56,28)(29,93,81,41)(30,94,82,42)(31,95,83,43)(32,96,84,44)(37,89,77,49)(38,90,78,50)(39,91,79,51)(40,92,80,52), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,81,39)(14,82,40)(15,83,37)(16,84,38)(17,87,21)(18,88,22)(19,85,23)(20,86,24)(25,93,51)(26,94,52)(27,95,49)(28,96,50)(29,79,53)(30,80,54)(31,77,55)(32,78,56)(33,63,59)(34,64,60)(35,61,57)(36,62,58)(41,91,65)(42,92,66)(43,89,67)(44,90,68)(45,75,71)(46,76,72)(47,73,69)(48,74,70), (2,54)(4,56)(5,9)(6,30)(7,11)(8,32)(10,80)(12,78)(14,36)(16,34)(17,23)(18,44)(19,21)(20,42)(22,90)(24,92)(25,27)(26,46)(28,48)(29,79)(31,77)(37,83)(38,64)(39,81)(40,62)(41,89)(43,91)(45,47)(49,93)(50,74)(51,95)(52,76)(57,61)(58,82)(59,63)(60,84)(65,67)(66,86)(68,88)(69,75)(70,96)(71,73)(72,94)(85,87)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,79)(6,80)(7,77)(8,78)(9,29)(10,30)(11,31)(12,32)(13,35)(14,36)(15,33)(16,34)(17,91)(18,92)(19,89)(20,90)(21,41)(22,42)(23,43)(24,44)(25,47)(26,48)(27,45)(28,46)(37,59)(38,60)(39,57)(40,58)(49,71)(50,72)(51,69)(52,70)(61,81)(62,82)(63,83)(64,84)(65,87)(66,88)(67,85)(68,86)(73,93)(74,94)(75,95)(76,96), (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), (1,47,35,87)(2,48,36,88)(3,45,33,85)(4,46,34,86)(5,69,57,17)(6,70,58,18)(7,71,59,19)(8,72,60,20)(9,73,61,21)(10,74,62,22)(11,75,63,23)(12,76,64,24)(13,65,53,25)(14,66,54,26)(15,67,55,27)(16,68,56,28)(29,93,81,41)(30,94,82,42)(31,95,83,43)(32,96,84,44)(37,89,77,49)(38,90,78,50)(39,91,79,51)(40,92,80,52), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,81,39)(14,82,40)(15,83,37)(16,84,38)(17,87,21)(18,88,22)(19,85,23)(20,86,24)(25,93,51)(26,94,52)(27,95,49)(28,96,50)(29,79,53)(30,80,54)(31,77,55)(32,78,56)(33,63,59)(34,64,60)(35,61,57)(36,62,58)(41,91,65)(42,92,66)(43,89,67)(44,90,68)(45,75,71)(46,76,72)(47,73,69)(48,74,70), (2,54)(4,56)(5,9)(6,30)(7,11)(8,32)(10,80)(12,78)(14,36)(16,34)(17,23)(18,44)(19,21)(20,42)(22,90)(24,92)(25,27)(26,46)(28,48)(29,79)(31,77)(37,83)(38,64)(39,81)(40,62)(41,89)(43,91)(45,47)(49,93)(50,74)(51,95)(52,76)(57,61)(58,82)(59,63)(60,84)(65,67)(66,86)(68,88)(69,75)(70,96)(71,73)(72,94)(85,87) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,79),(6,80),(7,77),(8,78),(9,29),(10,30),(11,31),(12,32),(13,35),(14,36),(15,33),(16,34),(17,91),(18,92),(19,89),(20,90),(21,41),(22,42),(23,43),(24,44),(25,47),(26,48),(27,45),(28,46),(37,59),(38,60),(39,57),(40,58),(49,71),(50,72),(51,69),(52,70),(61,81),(62,82),(63,83),(64,84),(65,87),(66,88),(67,85),(68,86),(73,93),(74,94),(75,95),(76,96)], [(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)], [(1,47,35,87),(2,48,36,88),(3,45,33,85),(4,46,34,86),(5,69,57,17),(6,70,58,18),(7,71,59,19),(8,72,60,20),(9,73,61,21),(10,74,62,22),(11,75,63,23),(12,76,64,24),(13,65,53,25),(14,66,54,26),(15,67,55,27),(16,68,56,28),(29,93,81,41),(30,94,82,42),(31,95,83,43),(32,96,84,44),(37,89,77,49),(38,90,78,50),(39,91,79,51),(40,92,80,52)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,81,39),(14,82,40),(15,83,37),(16,84,38),(17,87,21),(18,88,22),(19,85,23),(20,86,24),(25,93,51),(26,94,52),(27,95,49),(28,96,50),(29,79,53),(30,80,54),(31,77,55),(32,78,56),(33,63,59),(34,64,60),(35,61,57),(36,62,58),(41,91,65),(42,92,66),(43,89,67),(44,90,68),(45,75,71),(46,76,72),(47,73,69),(48,74,70)], [(2,54),(4,56),(5,9),(6,30),(7,11),(8,32),(10,80),(12,78),(14,36),(16,34),(17,23),(18,44),(19,21),(20,42),(22,90),(24,92),(25,27),(26,46),(28,48),(29,79),(31,77),(37,83),(38,64),(39,81),(40,62),(41,89),(43,91),(45,47),(49,93),(50,74),(51,95),(52,76),(57,61),(58,82),(59,63),(60,84),(65,67),(66,86),(68,88),(69,75),(70,96),(71,73),(72,94),(85,87)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | ··· | 4L | 4M | ··· | 4R | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | C4○D12 |
| kernel | (C2×C42)⋊3S3 | C6.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | C2×C4×C12 | D6⋊C4 | C2×C42 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 1 | 4 | 3 | 8 | 4 | 4 | 4 | 16 |
Matrix representation of (C2×C42)⋊3S3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 4 | 2 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,10,6,0,0,0,7,3,0,0,0,0,0,11,4,0,0,0,9,2],[5,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,12,0],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12] >;
(C2×C42)⋊3S3 in GAP, Magma, Sage, TeX
(C_2\times C_4^2)\rtimes_3S_3
% in TeX
G:=Group("(C2xC4^2):3S3"); // GroupNames label
G:=SmallGroup(192,499);
// by ID
G=gap.SmallGroup(192,499);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,758,58,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations