metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊S3⋊2C4, D4⋊2(C4×S3), D12⋊3(C2×C4), C6.33(C4×D4), C3⋊2(D8⋊C4), C4⋊C4.139D6, (D4×Dic3)⋊2C2, C24⋊C4⋊18C2, (C2×C8).171D6, Dic3⋊5D4⋊2C2, D4⋊C4⋊20S3, (C2×D4).138D6, C2.D24⋊26C2, C2.4(D8⋊S3), C2.2(Q8⋊3D6), C12.8(C22×C4), C12.Q8⋊8C2, C22.72(S3×D4), C6.57(C8⋊C22), (C6×D4).46C22, C12.153(C4○D4), C4.50(D4⋊2S3), (C2×C12).225C23, (C2×C24).231C22, (C2×Dic3).145D4, (C2×D12).53C22, C4⋊Dic3.77C22, (C4×Dic3).12C22, C2.17(Dic3⋊4D4), C3⋊C8⋊2(C2×C4), C4.8(S3×C2×C4), (C3×D4)⋊4(C2×C4), (C2×D4⋊S3).2C2, (C2×C6).238(C2×D4), (C2×C3⋊C8).23C22, (C3×D4⋊C4)⋊31C2, (C3×C4⋊C4).26C22, (C2×C4).332(C22×S3), SmallGroup(192,344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊S3⋊C4
G = < a,b,c,d,e | a4=b2=c3=d2=e4=1, bab=dad=eae-1=a-1, ac=ca, bc=cb, dbd=ab, ebe-1=a-1b, dcd=c-1, ce=ec, de=ed >
Subgroups: 408 in 132 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C8⋊C4, D4⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, D4⋊S3, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C22×Dic3, C6×D4, D8⋊C4, C12.Q8, C24⋊C4, C2.D24, C3×D4⋊C4, Dic3⋊5D4, C2×D4⋊S3, D4×Dic3, D4⋊S3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8⋊C22, S3×C2×C4, S3×D4, D4⋊2S3, D8⋊C4, Dic3⋊4D4, D8⋊S3, Q8⋊3D6, D4⋊S3⋊C4
►On 96 points
Generators in S96
(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 69)(2 72)(3 71)(4 70)(5 43)(6 42)(7 41)(8 44)(9 39)(10 38)(11 37)(12 40)(13 76)(14 75)(15 74)(16 73)(17 78)(18 77)(19 80)(20 79)(21 58)(22 57)(23 60)(24 59)(25 64)(26 63)(27 62)(28 61)(29 66)(30 65)(31 68)(32 67)(33 94)(34 93)(35 96)(36 95)(45 82)(46 81)(47 84)(48 83)(49 91)(50 90)(51 89)(52 92)(53 87)(54 86)(55 85)(56 88)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(1 22)(2 21)(3 24)(4 23)(5 89)(6 92)(7 91)(8 90)(9 85)(10 88)(11 87)(12 86)(13 25)(14 28)(15 27)(16 26)(17 31)(18 30)(19 29)(20 32)(33 47)(34 46)(35 45)(36 48)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(57 72)(58 71)(59 70)(60 69)(61 74)(62 73)(63 76)(64 75)(65 80)(66 79)(67 78)(68 77)(81 96)(82 95)(83 94)(84 93)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 76 85 67)(6 75 86 66)(7 74 87 65)(8 73 88 68)(9 78 89 63)(10 77 90 62)(11 80 91 61)(12 79 92 64)(13 56 32 44)(14 55 29 43)(15 54 30 42)(16 53 31 41)(17 52 26 40)(18 51 27 39)(19 50 28 38)(20 49 25 37)(57 94 69 82)(58 93 70 81)(59 96 71 84)(60 95 72 83)
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), (1,69)(2,72)(3,71)(4,70)(5,43)(6,42)(7,41)(8,44)(9,39)(10,38)(11,37)(12,40)(13,76)(14,75)(15,74)(16,73)(17,78)(18,77)(19,80)(20,79)(21,58)(22,57)(23,60)(24,59)(25,64)(26,63)(27,62)(28,61)(29,66)(30,65)(31,68)(32,67)(33,94)(34,93)(35,96)(36,95)(45,82)(46,81)(47,84)(48,83)(49,91)(50,90)(51,89)(52,92)(53,87)(54,86)(55,85)(56,88), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (1,22)(2,21)(3,24)(4,23)(5,89)(6,92)(7,91)(8,90)(9,85)(10,88)(11,87)(12,86)(13,25)(14,28)(15,27)(16,26)(17,31)(18,30)(19,29)(20,32)(33,47)(34,46)(35,45)(36,48)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(57,72)(58,71)(59,70)(60,69)(61,74)(62,73)(63,76)(64,75)(65,80)(66,79)(67,78)(68,77)(81,96)(82,95)(83,94)(84,93), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83)>;
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), (1,69)(2,72)(3,71)(4,70)(5,43)(6,42)(7,41)(8,44)(9,39)(10,38)(11,37)(12,40)(13,76)(14,75)(15,74)(16,73)(17,78)(18,77)(19,80)(20,79)(21,58)(22,57)(23,60)(24,59)(25,64)(26,63)(27,62)(28,61)(29,66)(30,65)(31,68)(32,67)(33,94)(34,93)(35,96)(36,95)(45,82)(46,81)(47,84)(48,83)(49,91)(50,90)(51,89)(52,92)(53,87)(54,86)(55,85)(56,88), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (1,22)(2,21)(3,24)(4,23)(5,89)(6,92)(7,91)(8,90)(9,85)(10,88)(11,87)(12,86)(13,25)(14,28)(15,27)(16,26)(17,31)(18,30)(19,29)(20,32)(33,47)(34,46)(35,45)(36,48)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(57,72)(58,71)(59,70)(60,69)(61,74)(62,73)(63,76)(64,75)(65,80)(66,79)(67,78)(68,77)(81,96)(82,95)(83,94)(84,93), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83) );
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)], [(1,69),(2,72),(3,71),(4,70),(5,43),(6,42),(7,41),(8,44),(9,39),(10,38),(11,37),(12,40),(13,76),(14,75),(15,74),(16,73),(17,78),(18,77),(19,80),(20,79),(21,58),(22,57),(23,60),(24,59),(25,64),(26,63),(27,62),(28,61),(29,66),(30,65),(31,68),(32,67),(33,94),(34,93),(35,96),(36,95),(45,82),(46,81),(47,84),(48,83),(49,91),(50,90),(51,89),(52,92),(53,87),(54,86),(55,85),(56,88)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,95,10),(6,96,11),(7,93,12),(8,94,9),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,39,44),(34,40,41),(35,37,42),(36,38,43),(45,51,56),(46,52,53),(47,49,54),(48,50,55),(57,63,68),(58,64,65),(59,61,66),(60,62,67),(69,78,73),(70,79,74),(71,80,75),(72,77,76),(81,92,87),(82,89,88),(83,90,85),(84,91,86)], [(1,22),(2,21),(3,24),(4,23),(5,89),(6,92),(7,91),(8,90),(9,85),(10,88),(11,87),(12,86),(13,25),(14,28),(15,27),(16,26),(17,31),(18,30),(19,29),(20,32),(33,47),(34,46),(35,45),(36,48),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(57,72),(58,71),(59,70),(60,69),(61,74),(62,73),(63,76),(64,75),(65,80),(66,79),(67,78),(68,77),(81,96),(82,95),(83,94),(84,93)], [(1,46,22,34),(2,45,23,33),(3,48,24,36),(4,47,21,35),(5,76,85,67),(6,75,86,66),(7,74,87,65),(8,73,88,68),(9,78,89,63),(10,77,90,62),(11,80,91,61),(12,79,92,64),(13,56,32,44),(14,55,29,43),(15,54,30,42),(16,53,31,41),(17,52,26,40),(18,51,27,39),(19,50,28,38),(20,49,25,37),(57,94,69,82),(58,93,70,81),(59,96,71,84),(60,95,72,83)]])
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 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | C8⋊C22 | D4⋊2S3 | S3×D4 | D8⋊S3 | Q8⋊3D6 |
| kernel | D4⋊S3⋊C4 | C12.Q8 | C24⋊C4 | C2.D24 | C3×D4⋊C4 | Dic3⋊5D4 | C2×D4⋊S3 | D4×Dic3 | D4⋊S3 | D4⋊C4 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of D4⋊S3⋊C4 ►in GL6(𝔽73)
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 50 | 47 | 47 |
| 0 | 0 | 50 | 50 | 26 | 47 |
| 0 | 0 | 47 | 26 | 50 | 50 |
| 0 | 0 | 47 | 47 | 50 | 23 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 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 | 1 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
G:=sub<GL(6,GF(73))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,50,47,47,0,0,50,50,26,47,0,0,47,26,50,50,0,0,47,47,50,23],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
D4⋊S3⋊C4 in GAP, Magma, Sage, TeX
D_4\rtimes S_3\rtimes C_4
% in TeX
G:=Group("D4:S3:C4"); // GroupNames label
G:=SmallGroup(192,344);
// by ID
G=gap.SmallGroup(192,344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,219,58,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^2=e^4=1,b*a*b=d*a*d=e*a*e^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,e*b*e^-1=a^-1*b,d*c*d=c^-1,c*e=e*c,d*e=e*d>;
// generators/relations