direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C22×C6, C25⋊6C6, C12⋊4C24, C6.21C25, C4⋊(C23×C6), (C2×C6)⋊2C24, (C24×C6)⋊2C2, C24⋊12(C2×C6), (C23×C4)⋊11C6, C2.1(C24×C6), (C2×C12)⋊17C23, (C23×C12)⋊16C2, C22⋊2(C23×C6), C23⋊4(C22×C6), (C22×C6)⋊8C23, (C23×C6)⋊17C22, (C22×C12)⋊66C22, (C2×C4)⋊4(C22×C6), (C22×C4)⋊21(C2×C6), SmallGroup(192,1531)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1874 in 1362 conjugacy classes, 850 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, C23, C23, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C24, C24, C2×C12, C3×D4, C22×C6, C22×C6, C23×C4, C22×D4, C25, C22×C12, C6×D4, C23×C6, C23×C6, C23×C6, D4×C23, C23×C12, D4×C2×C6, C24×C6, D4×C22×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C25, C6×D4, C23×C6, D4×C23, D4×C2×C6, C24×C6, D4×C22×C6
►On 96 points
Generators in S96
(1 54)(2 49)(3 50)(4 51)(5 52)(6 53)(7 60)(8 55)(9 56)(10 57)(11 58)(12 59)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 10 15 96)(2 11 16 91)(3 12 17 92)(4 7 18 93)(5 8 13 94)(6 9 14 95)(19 88 28 79)(20 89 29 80)(21 90 30 81)(22 85 25 82)(23 86 26 83)(24 87 27 84)(31 76 40 67)(32 77 41 68)(33 78 42 69)(34 73 37 70)(35 74 38 71)(36 75 39 72)(43 64 52 55)(44 65 53 56)(45 66 54 57)(46 61 49 58)(47 62 50 59)(48 63 51 60)
(1 84)(2 79)(3 80)(4 81)(5 82)(6 83)(7 30)(8 25)(9 26)(10 27)(11 28)(12 29)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)
G:=sub<Sym(96)| (1,54)(2,49)(3,50)(4,51)(5,52)(6,53)(7,60)(8,55)(9,56)(10,57)(11,58)(12,59)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,10,15,96)(2,11,16,91)(3,12,17,92)(4,7,18,93)(5,8,13,94)(6,9,14,95)(19,88,28,79)(20,89,29,80)(21,90,30,81)(22,85,25,82)(23,86,26,83)(24,87,27,84)(31,76,40,67)(32,77,41,68)(33,78,42,69)(34,73,37,70)(35,74,38,71)(36,75,39,72)(43,64,52,55)(44,65,53,56)(45,66,54,57)(46,61,49,58)(47,62,50,59)(48,63,51,60), (1,84)(2,79)(3,80)(4,81)(5,82)(6,83)(7,30)(8,25)(9,26)(10,27)(11,28)(12,29)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)>;
G:=Group( (1,54)(2,49)(3,50)(4,51)(5,52)(6,53)(7,60)(8,55)(9,56)(10,57)(11,58)(12,59)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,10,15,96)(2,11,16,91)(3,12,17,92)(4,7,18,93)(5,8,13,94)(6,9,14,95)(19,88,28,79)(20,89,29,80)(21,90,30,81)(22,85,25,82)(23,86,26,83)(24,87,27,84)(31,76,40,67)(32,77,41,68)(33,78,42,69)(34,73,37,70)(35,74,38,71)(36,75,39,72)(43,64,52,55)(44,65,53,56)(45,66,54,57)(46,61,49,58)(47,62,50,59)(48,63,51,60), (1,84)(2,79)(3,80)(4,81)(5,82)(6,83)(7,30)(8,25)(9,26)(10,27)(11,28)(12,29)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78) );
G=PermutationGroup([[(1,54),(2,49),(3,50),(4,51),(5,52),(6,53),(7,60),(8,55),(9,56),(10,57),(11,58),(12,59),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(1,24),(2,19),(3,20),(4,21),(5,22),(6,23),(7,90),(8,85),(9,86),(10,87),(11,88),(12,89),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,10,15,96),(2,11,16,91),(3,12,17,92),(4,7,18,93),(5,8,13,94),(6,9,14,95),(19,88,28,79),(20,89,29,80),(21,90,30,81),(22,85,25,82),(23,86,26,83),(24,87,27,84),(31,76,40,67),(32,77,41,68),(33,78,42,69),(34,73,37,70),(35,74,38,71),(36,75,39,72),(43,64,52,55),(44,65,53,56),(45,66,54,57),(46,61,49,58),(47,62,50,59),(48,63,51,60)], [(1,84),(2,79),(3,80),(4,81),(5,82),(6,83),(7,30),(8,25),(9,26),(10,27),(11,28),(12,29),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2AE | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6AD | 6AE | ··· | 6BJ | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 |
| kernel | D4×C22×C6 | C23×C12 | D4×C2×C6 | C24×C6 | D4×C23 | C23×C4 | C22×D4 | C25 | C22×C6 | C23 |
| # reps | 1 | 1 | 28 | 2 | 2 | 2 | 56 | 4 | 8 | 16 |
Matrix representation of D4×C22×C6 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,10,0,0,0,0,0,10],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,0],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
D4×C22×C6 in GAP, Magma, Sage, TeX
D_4\times C_2^2\times C_6
% in TeX
G:=Group("D4xC2^2xC6"); // GroupNames label
G:=SmallGroup(192,1531);
// by ID
G=gap.SmallGroup(192,1531);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations