direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C4≀C2, C4○D4⋊3C28, D4⋊5(C2×C28), (C2×D4)⋊9C28, (C2×Q8)⋊7C28, Q8⋊5(C2×C28), (D4×C14)⋊21C4, (C2×C42)⋊6C14, (Q8×C14)⋊17C4, C4.72(D4×C14), (C4×C28)⋊56C22, C42⋊15(C2×C14), (C2×C28).519D4, C28.477(C2×D4), C4.7(C22×C28), C23.39(C7×D4), M4(2)⋊9(C2×C14), C22.12(D4×C14), (C14×M4(2))⋊30C2, (C2×M4(2))⋊12C14, C28.152(C22×C4), (C2×C28).896C23, (C22×C14).161D4, C28.117(C22⋊C4), (C7×M4(2))⋊38C22, (C22×C28).584C22, (C2×C4×C28)⋊19C2, (C7×C4○D4)⋊9C4, (C7×D4)⋊25(C2×C4), (C7×Q8)⋊23(C2×C4), (C2×C4).70(C7×D4), (C2×C4).50(C2×C28), C4○D4.6(C2×C14), (C2×C4○D4).6C14, C4.33(C7×C22⋊C4), (C2×C28).271(C2×C4), (C14×C4○D4).20C2, (C2×C14).407(C2×D4), C2.23(C14×C22⋊C4), C22.6(C7×C22⋊C4), C14.111(C2×C22⋊C4), (C2×C4).71(C22×C14), (C7×C4○D4).51C22, (C22×C4).113(C2×C14), (C2×C14).140(C22⋊C4), SmallGroup(448,828)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C4≀C2
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C4≀C2, C4×C28, C4×C28, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4≀C2, C2×C4×C28, C14×M4(2), C14×C4○D4, C14×C4≀C2
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C4≀C2, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C2×C4≀C2, C7×C22⋊C4, C22×C28, D4×C14, C7×C4≀C2, C14×C22⋊C4, C14×C4≀C2
►On 112 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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 75 16 86)(2 76 17 87)(3 77 18 88)(4 78 19 89)(5 79 20 90)(6 80 21 91)(7 81 22 92)(8 82 23 93)(9 83 24 94)(10 84 25 95)(11 71 26 96)(12 72 27 97)(13 73 28 98)(14 74 15 85)(29 111 59 44)(30 112 60 45)(31 99 61 46)(32 100 62 47)(33 101 63 48)(34 102 64 49)(35 103 65 50)(36 104 66 51)(37 105 67 52)(38 106 68 53)(39 107 69 54)(40 108 70 55)(41 109 57 56)(42 110 58 43)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 99)(13 100)(14 101)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 43)(25 44)(26 45)(27 46)(28 47)(29 84)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)
(1 82 16 93)(2 83 17 94)(3 84 18 95)(4 71 19 96)(5 72 20 97)(6 73 21 98)(7 74 22 85)(8 75 23 86)(9 76 24 87)(10 77 25 88)(11 78 26 89)(12 79 27 90)(13 80 28 91)(14 81 15 92)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 99)(54 100)(55 101)(56 102)
G:=sub<Sym(112)| (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), (1,75,16,86)(2,76,17,87)(3,77,18,88)(4,78,19,89)(5,79,20,90)(6,80,21,91)(7,81,22,92)(8,82,23,93)(9,83,24,94)(10,84,25,95)(11,71,26,96)(12,72,27,97)(13,73,28,98)(14,74,15,85)(29,111,59,44)(30,112,60,45)(31,99,61,46)(32,100,62,47)(33,101,63,48)(34,102,64,49)(35,103,65,50)(36,104,66,51)(37,105,67,52)(38,106,68,53)(39,107,69,54)(40,108,70,55)(41,109,57,56)(42,110,58,43), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,99)(13,100)(14,101)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47)(29,84)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92), (1,82,16,93)(2,83,17,94)(3,84,18,95)(4,71,19,96)(5,72,20,97)(6,73,21,98)(7,74,22,85)(8,75,23,86)(9,76,24,87)(10,77,25,88)(11,78,26,89)(12,79,27,90)(13,80,28,91)(14,81,15,92)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,99)(54,100)(55,101)(56,102)>;
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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,75,16,86)(2,76,17,87)(3,77,18,88)(4,78,19,89)(5,79,20,90)(6,80,21,91)(7,81,22,92)(8,82,23,93)(9,83,24,94)(10,84,25,95)(11,71,26,96)(12,72,27,97)(13,73,28,98)(14,74,15,85)(29,111,59,44)(30,112,60,45)(31,99,61,46)(32,100,62,47)(33,101,63,48)(34,102,64,49)(35,103,65,50)(36,104,66,51)(37,105,67,52)(38,106,68,53)(39,107,69,54)(40,108,70,55)(41,109,57,56)(42,110,58,43), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,99)(13,100)(14,101)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47)(29,84)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92), (1,82,16,93)(2,83,17,94)(3,84,18,95)(4,71,19,96)(5,72,20,97)(6,73,21,98)(7,74,22,85)(8,75,23,86)(9,76,24,87)(10,77,25,88)(11,78,26,89)(12,79,27,90)(13,80,28,91)(14,81,15,92)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,99)(54,100)(55,101)(56,102) );
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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,75,16,86),(2,76,17,87),(3,77,18,88),(4,78,19,89),(5,79,20,90),(6,80,21,91),(7,81,22,92),(8,82,23,93),(9,83,24,94),(10,84,25,95),(11,71,26,96),(12,72,27,97),(13,73,28,98),(14,74,15,85),(29,111,59,44),(30,112,60,45),(31,99,61,46),(32,100,62,47),(33,101,63,48),(34,102,64,49),(35,103,65,50),(36,104,66,51),(37,105,67,52),(38,106,68,53),(39,107,69,54),(40,108,70,55),(41,109,57,56),(42,110,58,43)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,99),(13,100),(14,101),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,43),(25,44),(26,45),(27,46),(28,47),(29,84),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92)], [(1,82,16,93),(2,83,17,94),(3,84,18,95),(4,71,19,96),(5,72,20,97),(6,73,21,98),(7,74,22,85),(8,75,23,86),(9,76,24,87),(10,77,25,88),(11,78,26,89),(12,79,27,90),(13,80,28,91),(14,81,15,92),(29,66),(30,67),(31,68),(32,69),(33,70),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,99),(54,100),(55,101),(56,102)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28CF | 28CG | ··· | 28CR | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | D4 | D4 | C4≀C2 | C7×D4 | C7×D4 | C7×C4≀C2 |
| kernel | C14×C4≀C2 | C7×C4≀C2 | C2×C4×C28 | C14×M4(2) | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C4≀C2 | C4≀C2 | C2×C42 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2×C28 | C22×C14 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 24 | 6 | 6 | 6 | 12 | 12 | 24 | 3 | 1 | 8 | 18 | 6 | 48 |
Matrix representation of C14×C4≀C2 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 98 |
| 112 | 0 | 0 |
| 0 | 0 | 98 |
| 0 | 15 | 0 |
| 1 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(113))| [112,0,0,0,64,0,0,0,64],[1,0,0,0,15,0,0,0,98],[112,0,0,0,0,15,0,98,0],[1,0,0,0,98,0,0,0,1] >;
C14×C4≀C2 in GAP, Magma, Sage, TeX
C_{14}\times C_4\wr C_2 % in TeX
G:=Group("C14xC4wrC2"); // GroupNames label
G:=SmallGroup(448,828);
// by ID
G=gap.SmallGroup(448,828);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,9804,4911,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations