direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C14, C28⋊4C23, C24⋊3C14, C14.16C24, C4⋊(C22×C14), (C2×C14)⋊2C23, C23⋊4(C2×C14), (C23×C14)⋊2C2, (C22×C4)⋊5C14, C22⋊(C22×C14), (C2×C28)⋊15C22, (C22×C28)⋊12C2, C2.1(C23×C14), (C22×C14)⋊6C22, (C2×C4)⋊4(C2×C14), SmallGroup(224,190)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C2×C14
G = < a,b,c,d | a2=b14=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22×C4, C2×D4, C24, C28, C2×C14, C2×C14, C22×D4, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C22×C28, D4×C14, C23×C14, D4×C2×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D4×C14, C23×C14, D4×C2×C14
►On 112 points
Generators in S112
(1 52)(2 53)(3 54)(4 55)(5 56)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 70)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 110)(30 111)(31 112)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 85)(82 86)(83 87)(84 88)
(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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 75 69 36)(2 76 70 37)(3 77 57 38)(4 78 58 39)(5 79 59 40)(6 80 60 41)(7 81 61 42)(8 82 62 29)(9 83 63 30)(10 84 64 31)(11 71 65 32)(12 72 66 33)(13 73 67 34)(14 74 68 35)(15 104 53 94)(16 105 54 95)(17 106 55 96)(18 107 56 97)(19 108 43 98)(20 109 44 85)(21 110 45 86)(22 111 46 87)(23 112 47 88)(24 99 48 89)(25 100 49 90)(26 101 50 91)(27 102 51 92)(28 103 52 93)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 94)(16 95)(17 96)(18 97)(19 98)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(43 108)(44 109)(45 110)(46 111)(47 112)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)
G:=sub<Sym(112)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,70)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,110)(30,111)(31,112)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,85)(82,86)(83,87)(84,88), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,75,69,36)(2,76,70,37)(3,77,57,38)(4,78,58,39)(5,79,59,40)(6,80,60,41)(7,81,61,42)(8,82,62,29)(9,83,63,30)(10,84,64,31)(11,71,65,32)(12,72,66,33)(13,73,67,34)(14,74,68,35)(15,104,53,94)(16,105,54,95)(17,106,55,96)(18,107,56,97)(19,108,43,98)(20,109,44,85)(21,110,45,86)(22,111,46,87)(23,112,47,88)(24,99,48,89)(25,100,49,90)(26,101,50,91)(27,102,51,92)(28,103,52,93), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,94)(16,95)(17,96)(18,97)(19,98)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(43,108)(44,109)(45,110)(46,111)(47,112)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,70)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,110)(30,111)(31,112)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,85)(82,86)(83,87)(84,88), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,75,69,36)(2,76,70,37)(3,77,57,38)(4,78,58,39)(5,79,59,40)(6,80,60,41)(7,81,61,42)(8,82,62,29)(9,83,63,30)(10,84,64,31)(11,71,65,32)(12,72,66,33)(13,73,67,34)(14,74,68,35)(15,104,53,94)(16,105,54,95)(17,106,55,96)(18,107,56,97)(19,108,43,98)(20,109,44,85)(21,110,45,86)(22,111,46,87)(23,112,47,88)(24,99,48,89)(25,100,49,90)(26,101,50,91)(27,102,51,92)(28,103,52,93), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,94)(16,95)(17,96)(18,97)(19,98)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(43,108)(44,109)(45,110)(46,111)(47,112)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,70),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,110),(30,111),(31,112),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,85),(82,86),(83,87),(84,88)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,75,69,36),(2,76,70,37),(3,77,57,38),(4,78,58,39),(5,79,59,40),(6,80,60,41),(7,81,61,42),(8,82,62,29),(9,83,63,30),(10,84,64,31),(11,71,65,32),(12,72,66,33),(13,73,67,34),(14,74,68,35),(15,104,53,94),(16,105,54,95),(17,106,55,96),(18,107,56,97),(19,108,43,98),(20,109,44,85),(21,110,45,86),(22,111,46,87),(23,112,47,88),(24,99,48,89),(25,100,49,90),(26,101,50,91),(27,102,51,92),(28,103,52,93)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,94),(16,95),(17,96),(18,97),(19,98),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(43,108),(44,109),(45,110),(46,111),(47,112),(48,99),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76)]])
D4×C2×C14 is a maximal subgroup of
(D4×C14)⋊6C4 (C2×C14)⋊8D8 (C7×D4).31D4 C24.18D14 C24.19D14 C24.20D14 C24.21D14 C24.38D14 C24⋊7D14 C24.41D14 C24.42D14
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14CL | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | C7×D4 |
| kernel | D4×C2×C14 | C22×C28 | D4×C14 | C23×C14 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C14 | C22 |
| # reps | 1 | 1 | 12 | 2 | 6 | 6 | 72 | 12 | 4 | 24 |
Matrix representation of D4×C2×C14 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 2 |
| 0 | 0 | 26 | 11 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 18 | 2 |
| 0 | 0 | 27 | 11 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,18,26,0,0,2,11],[28,0,0,0,0,28,0,0,0,0,18,27,0,0,2,11] >;
D4×C2×C14 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_{14} % in TeX
G:=Group("D4xC2xC14"); // GroupNames label
G:=SmallGroup(224,190);
// by ID
G=gap.SmallGroup(224,190);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-7,-2,1369]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations