metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2S3⋊2C4, C4⋊C4.134D6, (C4×S3).38D4, D4.10(C4×S3), C4.157(S3×D4), (C2×C8).201D6, D4⋊C4⋊22S3, (C2×D4).133D6, C6.39(C4○D8), C12.106(C2×D4), C6.SD16⋊5C2, D4⋊Dic3⋊6C2, C12.7(C22×C4), Dic6.1(C2×C4), C22.71(S3×D4), C2.2(D8⋊3S3), D6.5(C22⋊C4), C2.Dic12⋊24C2, (C22×S3).46D4, (C6×D4).33C22, (C2×C12).212C23, (C2×C24).226C22, (C2×Dic3).201D4, C2.2(Q8.7D6), C3⋊1(C23.24D4), C4⋊Dic3.68C22, (C2×Dic6).56C22, Dic3.18(C22⋊C4), C4.7(S3×C2×C4), (S3×C2×C8)⋊17C2, C4⋊C4⋊7S3⋊2C2, (C3×D4).4(C2×C4), (C4×S3).13(C2×C4), (C2×C6).225(C2×D4), C6.19(C2×C22⋊C4), C2.20(S3×C22⋊C4), (C3×D4⋊C4)⋊26C2, (C2×D4⋊2S3).4C2, (C3×C4⋊C4).15C22, (C2×C3⋊C8).211C22, (S3×C2×C4).223C22, (C2×C4).319(C22×S3), SmallGroup(192,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2S3⋊C4
G = < a,b,c,d,e | a4=b2=c3=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 424 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, D4⋊C4, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, S3×C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, D4⋊2S3, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C23.24D4, C6.SD16, C2.Dic12, D4⋊Dic3, C3×D4⋊C4, C4⋊C4⋊7S3, S3×C2×C8, C2×D4⋊2S3, D4⋊2S3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C4○D8, S3×C2×C4, S3×D4, C23.24D4, S3×C22⋊C4, D8⋊3S3, Q8.7D6, D4⋊2S3⋊C4
►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 58)(2 57)(3 60)(4 59)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 71)(22 70)(23 69)(24 72)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 83)(34 82)(35 81)(36 84)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 95)(46 94)(47 93)(48 96)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 93 10)(6 94 11)(7 95 12)(8 96 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 44 39)(34 41 40)(35 42 37)(36 43 38)(45 49 54)(46 50 55)(47 51 56)(48 52 53)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 75 80)(70 76 77)(71 73 78)(72 74 79)(81 87 92)(82 88 89)(83 85 90)(84 86 91)
(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(25 30)(26 31)(27 32)(28 29)(37 42)(38 43)(39 44)(40 41)(49 54)(50 55)(51 56)(52 53)(57 59)(58 60)(61 68)(62 65)(63 66)(64 67)(69 71)(70 72)(73 80)(74 77)(75 78)(76 79)(81 83)(82 84)(85 92)(86 89)(87 90)(88 91)(93 95)(94 96)
(1 95 21 82)(2 94 22 81)(3 93 23 84)(4 96 24 83)(5 25 91 14)(6 28 92 13)(7 27 89 16)(8 26 90 15)(9 31 85 20)(10 30 86 19)(11 29 87 18)(12 32 88 17)(33 58 48 71)(34 57 45 70)(35 60 46 69)(36 59 47 72)(37 67 55 80)(38 66 56 79)(39 65 53 78)(40 68 54 77)(41 63 49 76)(42 62 50 75)(43 61 51 74)(44 64 52 73)
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), (1,58)(2,57)(3,60)(4,59)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,71)(22,70)(23,69)(24,72)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,83)(34,82)(35,81)(36,84)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,95)(46,94)(47,93)(48,96), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,93,10)(6,94,11)(7,95,12)(8,96,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,44,39)(34,41,40)(35,42,37)(36,43,38)(45,49,54)(46,50,55)(47,51,56)(48,52,53)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,75,80)(70,76,77)(71,73,78)(72,74,79)(81,87,92)(82,88,89)(83,85,90)(84,86,91), (5,12)(6,9)(7,10)(8,11)(13,18)(14,19)(15,20)(16,17)(25,30)(26,31)(27,32)(28,29)(37,42)(38,43)(39,44)(40,41)(49,54)(50,55)(51,56)(52,53)(57,59)(58,60)(61,68)(62,65)(63,66)(64,67)(69,71)(70,72)(73,80)(74,77)(75,78)(76,79)(81,83)(82,84)(85,92)(86,89)(87,90)(88,91)(93,95)(94,96), (1,95,21,82)(2,94,22,81)(3,93,23,84)(4,96,24,83)(5,25,91,14)(6,28,92,13)(7,27,89,16)(8,26,90,15)(9,31,85,20)(10,30,86,19)(11,29,87,18)(12,32,88,17)(33,58,48,71)(34,57,45,70)(35,60,46,69)(36,59,47,72)(37,67,55,80)(38,66,56,79)(39,65,53,78)(40,68,54,77)(41,63,49,76)(42,62,50,75)(43,61,51,74)(44,64,52,73)>;
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), (1,58)(2,57)(3,60)(4,59)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,71)(22,70)(23,69)(24,72)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,83)(34,82)(35,81)(36,84)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,95)(46,94)(47,93)(48,96), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,93,10)(6,94,11)(7,95,12)(8,96,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,44,39)(34,41,40)(35,42,37)(36,43,38)(45,49,54)(46,50,55)(47,51,56)(48,52,53)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,75,80)(70,76,77)(71,73,78)(72,74,79)(81,87,92)(82,88,89)(83,85,90)(84,86,91), (5,12)(6,9)(7,10)(8,11)(13,18)(14,19)(15,20)(16,17)(25,30)(26,31)(27,32)(28,29)(37,42)(38,43)(39,44)(40,41)(49,54)(50,55)(51,56)(52,53)(57,59)(58,60)(61,68)(62,65)(63,66)(64,67)(69,71)(70,72)(73,80)(74,77)(75,78)(76,79)(81,83)(82,84)(85,92)(86,89)(87,90)(88,91)(93,95)(94,96), (1,95,21,82)(2,94,22,81)(3,93,23,84)(4,96,24,83)(5,25,91,14)(6,28,92,13)(7,27,89,16)(8,26,90,15)(9,31,85,20)(10,30,86,19)(11,29,87,18)(12,32,88,17)(33,58,48,71)(34,57,45,70)(35,60,46,69)(36,59,47,72)(37,67,55,80)(38,66,56,79)(39,65,53,78)(40,68,54,77)(41,63,49,76)(42,62,50,75)(43,61,51,74)(44,64,52,73) );
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)], [(1,58),(2,57),(3,60),(4,59),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,71),(22,70),(23,69),(24,72),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,83),(34,82),(35,81),(36,84),(37,92),(38,91),(39,90),(40,89),(41,88),(42,87),(43,86),(44,85),(45,95),(46,94),(47,93),(48,96)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,93,10),(6,94,11),(7,95,12),(8,96,9),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,44,39),(34,41,40),(35,42,37),(36,43,38),(45,49,54),(46,50,55),(47,51,56),(48,52,53),(57,63,68),(58,64,65),(59,61,66),(60,62,67),(69,75,80),(70,76,77),(71,73,78),(72,74,79),(81,87,92),(82,88,89),(83,85,90),(84,86,91)], [(5,12),(6,9),(7,10),(8,11),(13,18),(14,19),(15,20),(16,17),(25,30),(26,31),(27,32),(28,29),(37,42),(38,43),(39,44),(40,41),(49,54),(50,55),(51,56),(52,53),(57,59),(58,60),(61,68),(62,65),(63,66),(64,67),(69,71),(70,72),(73,80),(74,77),(75,78),(76,79),(81,83),(82,84),(85,92),(86,89),(87,90),(88,91),(93,95),(94,96)], [(1,95,21,82),(2,94,22,81),(3,93,23,84),(4,96,24,83),(5,25,91,14),(6,28,92,13),(7,27,89,16),(8,26,90,15),(9,31,85,20),(10,30,86,19),(11,29,87,18),(12,32,88,17),(33,58,48,71),(34,57,45,70),(35,60,46,69),(36,59,47,72),(37,67,55,80),(38,66,56,79),(39,65,53,78),(40,68,54,77),(41,63,49,76),(42,62,50,75),(43,61,51,74),(44,64,52,73)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4×S3 | C4○D8 | S3×D4 | S3×D4 | D8⋊3S3 | Q8.7D6 |
| kernel | D4⋊2S3⋊C4 | C6.SD16 | C2.Dic12 | D4⋊Dic3 | C3×D4⋊C4 | C4⋊C4⋊7S3 | S3×C2×C8 | C2×D4⋊2S3 | D4⋊2S3 | D4⋊C4 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of D4⋊2S3⋊C4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 62 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 71 | 68 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 50 | 41 |
| 0 | 0 | 53 | 23 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,27,0,0,0,62,46],[1,0,0,0,0,1,0,0,0,0,5,71,0,0,12,68],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,1,0,0,0,5,72],[27,0,0,0,0,27,0,0,0,0,50,53,0,0,41,23] >;
D4⋊2S3⋊C4 in GAP, Magma, Sage, TeX
D_4\rtimes_2S_3\rtimes C_4
% in TeX
G:=Group("D4:2S3:C4"); // GroupNames label
G:=SmallGroup(192,331);
// by ID
G=gap.SmallGroup(192,331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,570,136,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^2=e^4=1,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations