direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.D6, SD16⋊9D6, C12.7C24, C24.35C23, Dic6.3C23, Dic12⋊17C22, C4.44(S3×D4), C3⋊C8.2C23, (C2×SD16)⋊5S3, (C6×SD16)⋊6C2, (C4×S3).16D4, D6.51(C2×D4), (C2×C8).103D6, C12.82(C2×D4), C4.7(S3×C23), (S3×Q8)⋊6C22, C8⋊S3⋊9C22, (C2×D4).183D6, C6⋊2(C8.C22), (C4×S3).4C23, C8.11(C22×S3), (C2×Q8).174D6, D4.5(C22×S3), C3⋊Q16⋊6C22, (C3×D4).5C23, (C2×Dic12)⋊26C2, (C3×Q8).1C23, D4.S3⋊10C22, Dic3.56(C2×D4), (C3×SD16)⋊9C22, (C22×S3).99D4, C6.108(C22×D4), C22.140(S3×D4), Q8.11(C22×S3), (C2×C24).117C22, (C2×C12).524C23, (C2×Dic3).193D4, (C6×D4).165C22, D4⋊2S3.4C22, (C6×Q8).147C22, (C2×Dic6).196C22, (C2×S3×Q8)⋊15C2, C2.81(C2×S3×D4), (C2×C8⋊S3)⋊5C2, C3⋊2(C2×C8.C22), (C2×D4.S3)⋊28C2, (C2×C3⋊Q16)⋊25C2, (C2×C6).397(C2×D4), (C2×C3⋊C8).180C22, (S3×C2×C4).157C22, (C2×D4⋊2S3).10C2, (C2×C4).613(C22×S3), SmallGroup(192,1319)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×8], C22, C22 [×8], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×2], Q8 [×11], C23 [×2], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C8, C2×C8, M4(2) [×4], SD16 [×4], SD16 [×4], Q16 [×8], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×9], C4○D4 [×6], C3⋊C8 [×2], C24 [×2], Dic6 [×4], Dic6 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×6], C3⋊D4 [×4], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C2×M4(2), C2×SD16, C2×SD16, C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C8⋊S3 [×4], Dic12 [×4], C2×C3⋊C8, D4.S3 [×4], C3⋊Q16 [×4], C2×C24, C3×SD16 [×4], C2×Dic6 [×2], C2×Dic6, S3×C2×C4, S3×C2×C4, D4⋊2S3 [×4], D4⋊2S3 [×2], S3×Q8 [×4], S3×Q8 [×2], C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C2×C8.C22, C2×C8⋊S3, C2×Dic12, D4.D6 [×8], C2×D4.S3, C2×C3⋊Q16, C6×SD16, C2×D4⋊2S3, C2×S3×Q8, C2×D4.D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C8.C22 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C8.C22, D4.D6 [×2], C2×S3×D4, C2×D4.D6
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >
►On 96 points
(1 56)(2 57)(3 58)(4 59)(5 60)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 84)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)
(1 78 7 84)(2 73 8 79)(3 80 9 74)(4 75 10 81)(5 82 11 76)(6 77 12 83)(13 61 19 67)(14 68 20 62)(15 63 21 69)(16 70 22 64)(17 65 23 71)(18 72 24 66)(25 56 31 50)(26 51 32 57)(27 58 33 52)(28 53 34 59)(29 60 35 54)(30 55 36 49)(37 95 43 89)(38 90 44 96)(39 85 45 91)(40 92 46 86)(41 87 47 93)(42 94 48 88)
(1 38)(2 85)(3 40)(4 87)(5 42)(6 89)(7 44)(8 91)(9 46)(10 93)(11 48)(12 95)(13 51)(14 27)(15 53)(16 29)(17 55)(18 31)(19 57)(20 33)(21 59)(22 35)(23 49)(24 25)(26 61)(28 63)(30 65)(32 67)(34 69)(36 71)(37 83)(39 73)(41 75)(43 77)(45 79)(47 81)(50 66)(52 68)(54 70)(56 72)(58 62)(60 64)(74 92)(76 94)(78 96)(80 86)(82 88)(84 90)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 44 43 38)(39 42 45 48)(40 47 46 41)(49 50 55 56)(51 60 57 54)(52 53 58 59)(61 64 67 70)(62 69 68 63)(65 72 71 66)(73 82 79 76)(74 75 80 81)(77 78 83 84)(85 88 91 94)(86 93 92 87)(89 96 95 90)
G:=sub<Sym(96)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,78,7,84)(2,73,8,79)(3,80,9,74)(4,75,10,81)(5,82,11,76)(6,77,12,83)(13,61,19,67)(14,68,20,62)(15,63,21,69)(16,70,22,64)(17,65,23,71)(18,72,24,66)(25,56,31,50)(26,51,32,57)(27,58,33,52)(28,53,34,59)(29,60,35,54)(30,55,36,49)(37,95,43,89)(38,90,44,96)(39,85,45,91)(40,92,46,86)(41,87,47,93)(42,94,48,88), (1,38)(2,85)(3,40)(4,87)(5,42)(6,89)(7,44)(8,91)(9,46)(10,93)(11,48)(12,95)(13,51)(14,27)(15,53)(16,29)(17,55)(18,31)(19,57)(20,33)(21,59)(22,35)(23,49)(24,25)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(37,83)(39,73)(41,75)(43,77)(45,79)(47,81)(50,66)(52,68)(54,70)(56,72)(58,62)(60,64)(74,92)(76,94)(78,96)(80,86)(82,88)(84,90), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,50,55,56)(51,60,57,54)(52,53,58,59)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,82,79,76)(74,75,80,81)(77,78,83,84)(85,88,91,94)(86,93,92,87)(89,96,95,90)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,78,7,84)(2,73,8,79)(3,80,9,74)(4,75,10,81)(5,82,11,76)(6,77,12,83)(13,61,19,67)(14,68,20,62)(15,63,21,69)(16,70,22,64)(17,65,23,71)(18,72,24,66)(25,56,31,50)(26,51,32,57)(27,58,33,52)(28,53,34,59)(29,60,35,54)(30,55,36,49)(37,95,43,89)(38,90,44,96)(39,85,45,91)(40,92,46,86)(41,87,47,93)(42,94,48,88), (1,38)(2,85)(3,40)(4,87)(5,42)(6,89)(7,44)(8,91)(9,46)(10,93)(11,48)(12,95)(13,51)(14,27)(15,53)(16,29)(17,55)(18,31)(19,57)(20,33)(21,59)(22,35)(23,49)(24,25)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(37,83)(39,73)(41,75)(43,77)(45,79)(47,81)(50,66)(52,68)(54,70)(56,72)(58,62)(60,64)(74,92)(76,94)(78,96)(80,86)(82,88)(84,90), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,50,55,56)(51,60,57,54)(52,53,58,59)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,82,79,76)(74,75,80,81)(77,78,83,84)(85,88,91,94)(86,93,92,87)(89,96,95,90) );
G=PermutationGroup([(1,56),(2,57),(3,58),(4,59),(5,60),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,84),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,81),(35,82),(36,83),(37,71),(38,72),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70)], [(1,78,7,84),(2,73,8,79),(3,80,9,74),(4,75,10,81),(5,82,11,76),(6,77,12,83),(13,61,19,67),(14,68,20,62),(15,63,21,69),(16,70,22,64),(17,65,23,71),(18,72,24,66),(25,56,31,50),(26,51,32,57),(27,58,33,52),(28,53,34,59),(29,60,35,54),(30,55,36,49),(37,95,43,89),(38,90,44,96),(39,85,45,91),(40,92,46,86),(41,87,47,93),(42,94,48,88)], [(1,38),(2,85),(3,40),(4,87),(5,42),(6,89),(7,44),(8,91),(9,46),(10,93),(11,48),(12,95),(13,51),(14,27),(15,53),(16,29),(17,55),(18,31),(19,57),(20,33),(21,59),(22,35),(23,49),(24,25),(26,61),(28,63),(30,65),(32,67),(34,69),(36,71),(37,83),(39,73),(41,75),(43,77),(45,79),(47,81),(50,66),(52,68),(54,70),(56,72),(58,62),(60,64),(74,92),(76,94),(78,96),(80,86),(82,88),(84,90)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,44,43,38),(39,42,45,48),(40,47,46,41),(49,50,55,56),(51,60,57,54),(52,53,58,59),(61,64,67,70),(62,69,68,63),(65,72,71,66),(73,82,79,76),(74,75,80,81),(77,78,83,84),(85,88,91,94),(86,93,92,87),(89,96,95,90)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 41 | 0 | 0 |
| 0 | 0 | 41 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 17 |
| 0 | 0 | 0 | 0 | 17 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 41 | 0 | 0 |
| 0 | 0 | 41 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 56 |
| 0 | 0 | 0 | 0 | 56 | 41 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,0,0,0,0,0,0,72,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,56,41,0,0,0,0,41,17,0,0,0,0,0,0,41,17,0,0,0,0,17,32],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,56,41,0,0,0,0,41,17,0,0,0,0,0,0,32,56,0,0,0,0,56,41] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 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 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | C8.C22 | S3×D4 | S3×D4 | D4.D6 |
| kernel | C2×D4.D6 | C2×C8⋊S3 | C2×Dic12 | D4.D6 | C2×D4.S3 | C2×C3⋊Q16 | C6×SD16 | C2×D4⋊2S3 | C2×S3×Q8 | C2×SD16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_4.D_6
% in TeX
G:=Group("C2xD4.D6"); // GroupNames label
G:=SmallGroup(192,1319);
// by ID
G=gap.SmallGroup(192,1319);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,1123,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations