metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14.5D4, C4⋊C4⋊2D7, C2.12(D4×D7), D14⋊C4⋊13C2, Dic7⋊C4⋊6C2, (C2×D28).3C2, (C2×C4).11D14, C14.25(C2×D4), C2.14(C4○D28), C14.12(C4○D4), (C2×C14).35C23, (C2×C28).57C22, C2.5(Q8⋊2D7), C7⋊3(C22.D4), (C22×D7).6C22, C22.49(C22×D7), (C2×Dic7).11C22, (C7×C4⋊C4)⋊5C2, (C2×C4×D7)⋊13C2, SmallGroup(224,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14.5D4
G = < a,b,c,d | a14=b2=c4=1, d2=a7, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a7b, dcd-1=c-1 >
Subgroups: 374 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, D7, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, D14, D14, C2×C14, C22.D4, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, D14.5D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C22×D7, C4○D28, D4×D7, Q8⋊2D7, D14.5D4
►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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(43 74)(44 73)(45 72)(46 71)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 85)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)
(1 112 63 50)(2 99 64 51)(3 100 65 52)(4 101 66 53)(5 102 67 54)(6 103 68 55)(7 104 69 56)(8 105 70 43)(9 106 57 44)(10 107 58 45)(11 108 59 46)(12 109 60 47)(13 110 61 48)(14 111 62 49)(15 93 74 42)(16 94 75 29)(17 95 76 30)(18 96 77 31)(19 97 78 32)(20 98 79 33)(21 85 80 34)(22 86 81 35)(23 87 82 36)(24 88 83 37)(25 89 84 38)(26 90 71 39)(27 91 72 40)(28 92 73 41)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 49 22 56)(16 50 23 43)(17 51 24 44)(18 52 25 45)(19 53 26 46)(20 54 27 47)(21 55 28 48)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(43,74)(44,73)(45,72)(46,71)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,85)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86), (1,112,63,50)(2,99,64,51)(3,100,65,52)(4,101,66,53)(5,102,67,54)(6,103,68,55)(7,104,69,56)(8,105,70,43)(9,106,57,44)(10,107,58,45)(11,108,59,46)(12,109,60,47)(13,110,61,48)(14,111,62,49)(15,93,74,42)(16,94,75,29)(17,95,76,30)(18,96,77,31)(19,97,78,32)(20,98,79,33)(21,85,80,34)(22,86,81,35)(23,87,82,36)(24,88,83,37)(25,89,84,38)(26,90,71,39)(27,91,72,40)(28,92,73,41), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,49,22,56)(16,50,23,43)(17,51,24,44)(18,52,25,45)(19,53,26,46)(20,54,27,47)(21,55,28,48)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107)>;
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(43,74)(44,73)(45,72)(46,71)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,85)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86), (1,112,63,50)(2,99,64,51)(3,100,65,52)(4,101,66,53)(5,102,67,54)(6,103,68,55)(7,104,69,56)(8,105,70,43)(9,106,57,44)(10,107,58,45)(11,108,59,46)(12,109,60,47)(13,110,61,48)(14,111,62,49)(15,93,74,42)(16,94,75,29)(17,95,76,30)(18,96,77,31)(19,97,78,32)(20,98,79,33)(21,85,80,34)(22,86,81,35)(23,87,82,36)(24,88,83,37)(25,89,84,38)(26,90,71,39)(27,91,72,40)(28,92,73,41), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,49,22,56)(16,50,23,43)(17,51,24,44)(18,52,25,45)(19,53,26,46)(20,54,27,47)(21,55,28,48)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107) );
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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(43,74),(44,73),(45,72),(46,71),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,85),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86)], [(1,112,63,50),(2,99,64,51),(3,100,65,52),(4,101,66,53),(5,102,67,54),(6,103,68,55),(7,104,69,56),(8,105,70,43),(9,106,57,44),(10,107,58,45),(11,108,59,46),(12,109,60,47),(13,110,61,48),(14,111,62,49),(15,93,74,42),(16,94,75,29),(17,95,76,30),(18,96,77,31),(19,97,78,32),(20,98,79,33),(21,85,80,34),(22,86,81,35),(23,87,82,36),(24,88,83,37),(25,89,84,38),(26,90,71,39),(27,91,72,40),(28,92,73,41)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,49,22,56),(16,50,23,43),(17,51,24,44),(18,52,25,45),(19,53,26,46),(20,54,27,47),(21,55,28,48),(57,95,64,88),(58,96,65,89),(59,97,66,90),(60,98,67,91),(61,85,68,92),(62,86,69,93),(63,87,70,94),(71,108,78,101),(72,109,79,102),(73,110,80,103),(74,111,81,104),(75,112,82,105),(76,99,83,106),(77,100,84,107)]])
D14.5D4 is a maximal subgroup of
C14.2- 1+4 C14.112+ 1+4 C14.62- 1+4 C42⋊10D14 C42.93D14 C42.100D14 C42.104D14 C42⋊12D14 D28⋊23D4 C42⋊17D14 C42.131D14 C42.132D14 C42.133D14 C42.136D14 C14.372+ 1+4 C14.402+ 1+4 C14.442+ 1+4 C14.482+ 1+4 C4⋊C4⋊26D14 C14.162- 1+4 D28⋊21D4 D28⋊22D4 C14.532+ 1+4 C14.202- 1+4 C14.222- 1+4 C14.562+ 1+4 C14.572+ 1+4 C14.262- 1+4 D7×C22.D4 C14.1202+ 1+4 C14.1212+ 1+4 C14.822- 1+4 C14.612+ 1+4 C14.662+ 1+4 C14.672+ 1+4 C14.862- 1+4 C42.237D14 C42.150D14 C42.151D14 C42.152D14 C42.153D14 C42.156D14 C42.157D14 C42.158D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.161D14 C42.163D14 C42.164D14 C42⋊25D14 C42.171D14 D28⋊12D4 C42.178D14 C42.180D14
D14.5D4 is a maximal quotient of
C14.(C4×D4) (C2×C4).Dic14 C22.58(D4×D7) (C2×C4)⋊9D28 D14⋊C4⋊5C4 (C2×Dic7)⋊3D4 (C22×D7).9D4 D14.2SD16 D14.4SD16 C4.Q8⋊D7 C28.(C4○D4) D14.5D8 D14.2Q16 C2.D8⋊D7 C2.D8⋊7D7 C4⋊C4⋊5Dic7 (C2×C28).54D4 D14⋊C4⋊6C4 D14⋊C4⋊7C4 (C2×C4)⋊3D28 (C2×C28).289D4 (C2×C28).290D4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | C4○D28 | D4×D7 | Q8⋊2D7 |
| kernel | D14.5D4 | Dic7⋊C4 | D14⋊C4 | C7×C4⋊C4 | C2×C4×D7 | C2×D28 | D14 | C4⋊C4 | C14 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 2 | 3 | 4 | 9 | 12 | 3 | 3 |
Matrix representation of D14.5D4 ►in GL4(𝔽29) generated by
| 25 | 19 | 0 | 0 |
| 15 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 22 | 25 | 0 | 0 |
| 12 | 7 | 0 | 0 |
| 0 | 0 | 5 | 26 |
| 0 | 0 | 8 | 24 |
| 11 | 15 | 0 | 0 |
| 21 | 18 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 11 | 17 |
| 13 | 23 | 0 | 0 |
| 9 | 16 | 0 | 0 |
| 0 | 0 | 24 | 3 |
| 0 | 0 | 1 | 5 |
G:=sub<GL(4,GF(29))| [25,15,0,0,19,1,0,0,0,0,28,0,0,0,0,28],[22,12,0,0,25,7,0,0,0,0,5,8,0,0,26,24],[11,21,0,0,15,18,0,0,0,0,12,11,0,0,0,17],[13,9,0,0,23,16,0,0,0,0,24,1,0,0,3,5] >;
D14.5D4 in GAP, Magma, Sage, TeX
D_{14}._5D_4 % in TeX
G:=Group("D14.5D4"); // GroupNames label
G:=SmallGroup(224,89);
// by ID
G=gap.SmallGroup(224,89);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,218,188,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations