direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D4⋊2S3, C42.227D6, D4⋊8(C4×S3), (C4×D4)⋊28S3, (D4×C12)⋊5C2, (S3×C42)⋊2C2, C4⋊C4.313D6, C12⋊15(C4○D4), Dic6⋊12(C2×C4), (C4×Dic6)⋊23C2, (D4×Dic3)⋊43C2, (C2×D4).240D6, (C2×C6).80C24, C6.20(C23×C4), D6.7(C22×C4), C12.31(C22×C4), C22⋊C4.128D6, (C22×C4).335D6, Dic3⋊11(C4○D4), Dic3⋊4D4⋊50C2, Dic6⋊C4⋊45C2, (C4×C12).143C22, (C2×C12).490C23, D6⋊C4.120C22, (C6×D4).247C22, C22.30(S3×C23), C23.16D6⋊31C2, C4⋊Dic3.361C22, (C22×C6).150C23, C23.171(C22×S3), Dic3.24(C22×C4), Dic3⋊C4.133C22, (C22×S3).167C23, (C22×C12).362C22, (C4×Dic3).291C22, (C2×Dic6).286C22, (C2×Dic3).307C23, C6.D4.100C22, (C22×Dic3).218C22, C3⋊3(C4×C4○D4), C4.31(S3×C2×C4), (C4×S3)⋊6(C2×C4), C3⋊D4⋊1(C2×C4), C2.3(S3×C4○D4), C22.1(S3×C2×C4), (C3×D4)⋊10(C2×C4), (C4×C3⋊D4)⋊37C2, C4⋊C4⋊7S3⋊46C2, (C2×C4×Dic3)⋊34C2, C2.22(S3×C22×C4), C6.135(C2×C4○D4), C2.5(C2×D4⋊2S3), (C2×C6).1(C22×C4), (C2×Dic3)⋊17(C2×C4), (S3×C2×C4).291C22, (C2×D4⋊2S3).11C2, (C3×C4⋊C4).316C22, (C2×C4).577(C22×S3), (C2×C3⋊D4).107C22, (C3×C22⋊C4).140C22, SmallGroup(192,1095)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D4⋊2S3
G = < a,b,c,d,e | a4=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 616 in 310 conjugacy classes, 157 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C4×C4○D4, C4×Dic6, S3×C42, C23.16D6, Dic3⋊4D4, Dic6⋊C4, C4⋊C4⋊7S3, C2×C4×Dic3, C4×C3⋊D4, D4×Dic3, D4×C12, C2×D4⋊2S3, C4×D4⋊2S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C23×C4, C2×C4○D4, S3×C2×C4, D4⋊2S3, S3×C23, C4×C4○D4, S3×C22×C4, C2×D4⋊2S3, S3×C4○D4, C4×D4⋊2S3
►On 96 points
(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 28 41 65)(2 25 42 66)(3 26 43 67)(4 27 44 68)(5 22 77 14)(6 23 78 15)(7 24 79 16)(8 21 80 13)(9 32 45 87)(10 29 46 88)(11 30 47 85)(12 31 48 86)(17 89 49 83)(18 90 50 84)(19 91 51 81)(20 92 52 82)(33 69 75 61)(34 70 76 62)(35 71 73 63)(36 72 74 64)(37 95 53 59)(38 96 54 60)(39 93 55 57)(40 94 56 58)
(1 83)(2 84)(3 81)(4 82)(5 76)(6 73)(7 74)(8 75)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 50)(26 51)(27 52)(28 49)(29 56)(30 53)(31 54)(32 55)(33 80)(34 77)(35 78)(36 79)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 68 86)(6 65 87)(7 66 88)(8 67 85)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 39 73)(18 40 74)(19 37 75)(20 38 76)(25 29 79)(26 30 80)(27 31 77)(28 32 78)(33 51 53)(34 52 54)(35 49 55)(36 50 56)(57 71 83)(58 72 84)(59 69 81)(60 70 82)(61 91 95)(62 92 96)(63 89 93)(64 90 94)
(1 41)(2 42)(3 43)(4 44)(5 31)(6 32)(7 29)(8 30)(9 15)(10 16)(11 13)(12 14)(21 47)(22 48)(23 45)(24 46)(25 66)(26 67)(27 68)(28 65)(33 53)(34 54)(35 55)(36 56)(37 75)(38 76)(39 73)(40 74)(57 71)(58 72)(59 69)(60 70)(61 95)(62 96)(63 93)(64 94)(77 86)(78 87)(79 88)(80 85)
G:=sub<Sym(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), (1,28,41,65)(2,25,42,66)(3,26,43,67)(4,27,44,68)(5,22,77,14)(6,23,78,15)(7,24,79,16)(8,21,80,13)(9,32,45,87)(10,29,46,88)(11,30,47,85)(12,31,48,86)(17,89,49,83)(18,90,50,84)(19,91,51,81)(20,92,52,82)(33,69,75,61)(34,70,76,62)(35,71,73,63)(36,72,74,64)(37,95,53,59)(38,96,54,60)(39,93,55,57)(40,94,56,58), (1,83)(2,84)(3,81)(4,82)(5,76)(6,73)(7,74)(8,75)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,50)(26,51)(27,52)(28,49)(29,56)(30,53)(31,54)(32,55)(33,80)(34,77)(35,78)(36,79)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,68,86)(6,65,87)(7,66,88)(8,67,85)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,39,73)(18,40,74)(19,37,75)(20,38,76)(25,29,79)(26,30,80)(27,31,77)(28,32,78)(33,51,53)(34,52,54)(35,49,55)(36,50,56)(57,71,83)(58,72,84)(59,69,81)(60,70,82)(61,91,95)(62,92,96)(63,89,93)(64,90,94), (1,41)(2,42)(3,43)(4,44)(5,31)(6,32)(7,29)(8,30)(9,15)(10,16)(11,13)(12,14)(21,47)(22,48)(23,45)(24,46)(25,66)(26,67)(27,68)(28,65)(33,53)(34,54)(35,55)(36,56)(37,75)(38,76)(39,73)(40,74)(57,71)(58,72)(59,69)(60,70)(61,95)(62,96)(63,93)(64,94)(77,86)(78,87)(79,88)(80,85)>;
G:=Group( (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), (1,28,41,65)(2,25,42,66)(3,26,43,67)(4,27,44,68)(5,22,77,14)(6,23,78,15)(7,24,79,16)(8,21,80,13)(9,32,45,87)(10,29,46,88)(11,30,47,85)(12,31,48,86)(17,89,49,83)(18,90,50,84)(19,91,51,81)(20,92,52,82)(33,69,75,61)(34,70,76,62)(35,71,73,63)(36,72,74,64)(37,95,53,59)(38,96,54,60)(39,93,55,57)(40,94,56,58), (1,83)(2,84)(3,81)(4,82)(5,76)(6,73)(7,74)(8,75)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,50)(26,51)(27,52)(28,49)(29,56)(30,53)(31,54)(32,55)(33,80)(34,77)(35,78)(36,79)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,68,86)(6,65,87)(7,66,88)(8,67,85)(13,43,47)(14,44,48)(15,41,45)(16,42,46)(17,39,73)(18,40,74)(19,37,75)(20,38,76)(25,29,79)(26,30,80)(27,31,77)(28,32,78)(33,51,53)(34,52,54)(35,49,55)(36,50,56)(57,71,83)(58,72,84)(59,69,81)(60,70,82)(61,91,95)(62,92,96)(63,89,93)(64,90,94), (1,41)(2,42)(3,43)(4,44)(5,31)(6,32)(7,29)(8,30)(9,15)(10,16)(11,13)(12,14)(21,47)(22,48)(23,45)(24,46)(25,66)(26,67)(27,68)(28,65)(33,53)(34,54)(35,55)(36,56)(37,75)(38,76)(39,73)(40,74)(57,71)(58,72)(59,69)(60,70)(61,95)(62,96)(63,93)(64,94)(77,86)(78,87)(79,88)(80,85) );
G=PermutationGroup([[(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)], [(1,28,41,65),(2,25,42,66),(3,26,43,67),(4,27,44,68),(5,22,77,14),(6,23,78,15),(7,24,79,16),(8,21,80,13),(9,32,45,87),(10,29,46,88),(11,30,47,85),(12,31,48,86),(17,89,49,83),(18,90,50,84),(19,91,51,81),(20,92,52,82),(33,69,75,61),(34,70,76,62),(35,71,73,63),(36,72,74,64),(37,95,53,59),(38,96,54,60),(39,93,55,57),(40,94,56,58)], [(1,83),(2,84),(3,81),(4,82),(5,76),(6,73),(7,74),(8,75),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,50),(26,51),(27,52),(28,49),(29,56),(30,53),(31,54),(32,55),(33,80),(34,77),(35,78),(36,79),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,9,23),(2,10,24),(3,11,21),(4,12,22),(5,68,86),(6,65,87),(7,66,88),(8,67,85),(13,43,47),(14,44,48),(15,41,45),(16,42,46),(17,39,73),(18,40,74),(19,37,75),(20,38,76),(25,29,79),(26,30,80),(27,31,77),(28,32,78),(33,51,53),(34,52,54),(35,49,55),(36,50,56),(57,71,83),(58,72,84),(59,69,81),(60,70,82),(61,91,95),(62,92,96),(63,89,93),(64,90,94)], [(1,41),(2,42),(3,43),(4,44),(5,31),(6,32),(7,29),(8,30),(9,15),(10,16),(11,13),(12,14),(21,47),(22,48),(23,45),(24,46),(25,66),(26,67),(27,68),(28,65),(33,53),(34,54),(35,55),(36,56),(37,75),(38,76),(39,73),(40,74),(57,71),(58,72),(59,69),(60,70),(61,95),(62,96),(63,93),(64,94),(77,86),(78,87),(79,88),(80,85)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AD | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4×S3 | D4⋊2S3 | S3×C4○D4 |
| kernel | C4×D4⋊2S3 | C4×Dic6 | S3×C42 | C23.16D6 | Dic3⋊4D4 | Dic6⋊C4 | C4⋊C4⋊7S3 | C2×C4×Dic3 | C4×C3⋊D4 | D4×Dic3 | D4×C12 | C2×D4⋊2S3 | D4⋊2S3 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 16 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C4×D4⋊2S3 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 5 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 8 | 0 | 0 |
| 10 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[5,0,0,0,8,8,0,0,0,0,12,0,0,0,0,12],[5,10,0,0,8,8,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[12,0,0,0,1,1,0,0,0,0,0,1,0,0,1,0] >;
C4×D4⋊2S3 in GAP, Magma, Sage, TeX
C_4\times D_4\rtimes_2S_3
% in TeX
G:=Group("C4xD4:2S3"); // GroupNames label
G:=SmallGroup(192,1095);
// by ID
G=gap.SmallGroup(192,1095);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,794,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations