metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D14, Q8.11D14, C7⋊22- 1+4, C14.13C24, C28.27C23, D14.8C23, D28.14C22, Dic7.8C23, Dic14.14C22, C4○D4⋊4D7, (Q8×D7)⋊5C2, C4○D28⋊9C2, C7⋊D4.C22, D4⋊2D7⋊5C2, (C2×C4).25D14, (C2×C14).5C23, (C4×D7).6C22, C2.14(C23×D7), C4.34(C22×D7), (C2×Dic14)⋊14C2, (C2×C28).49C22, (C7×D4).10C22, (C7×Q8).11C22, C22.4(C22×D7), (C2×Dic7).22C22, (C7×C4○D4)⋊5C2, SmallGroup(224,186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.10D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c13 >
Subgroups: 486 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C2×Dic14, C4○D28, D4⋊2D7, Q8×D7, C7×C4○D4, D4.10D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C22×D7, C23×D7, D4.10D14
►On 112 points
(1 89 15 103)(2 104 16 90)(3 91 17 105)(4 106 18 92)(5 93 19 107)(6 108 20 94)(7 95 21 109)(8 110 22 96)(9 97 23 111)(10 112 24 98)(11 99 25 85)(12 86 26 100)(13 101 27 87)(14 88 28 102)(29 81 43 67)(30 68 44 82)(31 83 45 69)(32 70 46 84)(33 57 47 71)(34 72 48 58)(35 59 49 73)(36 74 50 60)(37 61 51 75)(38 76 52 62)(39 63 53 77)(40 78 54 64)(41 65 55 79)(42 80 56 66)
(1 54)(2 41)(3 56)(4 43)(5 30)(6 45)(7 32)(8 47)(9 34)(10 49)(11 36)(12 51)(13 38)(14 53)(15 40)(16 55)(17 42)(18 29)(19 44)(20 31)(21 46)(22 33)(23 48)(24 35)(25 50)(26 37)(27 52)(28 39)(57 110)(58 97)(59 112)(60 99)(61 86)(62 101)(63 88)(64 103)(65 90)(66 105)(67 92)(68 107)(69 94)(70 109)(71 96)(72 111)(73 98)(74 85)(75 100)(76 87)(77 102)(78 89)(79 104)(80 91)(81 106)(82 93)(83 108)(84 95)
(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 70 15 84)(2 83 16 69)(3 68 17 82)(4 81 18 67)(5 66 19 80)(6 79 20 65)(7 64 21 78)(8 77 22 63)(9 62 23 76)(10 75 24 61)(11 60 25 74)(12 73 26 59)(13 58 27 72)(14 71 28 57)(29 92 43 106)(30 105 44 91)(31 90 45 104)(32 103 46 89)(33 88 47 102)(34 101 48 87)(35 86 49 100)(36 99 50 85)(37 112 51 98)(38 97 52 111)(39 110 53 96)(40 95 54 109)(41 108 55 94)(42 93 56 107)
G:=sub<Sym(112)| (1,89,15,103)(2,104,16,90)(3,91,17,105)(4,106,18,92)(5,93,19,107)(6,108,20,94)(7,95,21,109)(8,110,22,96)(9,97,23,111)(10,112,24,98)(11,99,25,85)(12,86,26,100)(13,101,27,87)(14,88,28,102)(29,81,43,67)(30,68,44,82)(31,83,45,69)(32,70,46,84)(33,57,47,71)(34,72,48,58)(35,59,49,73)(36,74,50,60)(37,61,51,75)(38,76,52,62)(39,63,53,77)(40,78,54,64)(41,65,55,79)(42,80,56,66), (1,54)(2,41)(3,56)(4,43)(5,30)(6,45)(7,32)(8,47)(9,34)(10,49)(11,36)(12,51)(13,38)(14,53)(15,40)(16,55)(17,42)(18,29)(19,44)(20,31)(21,46)(22,33)(23,48)(24,35)(25,50)(26,37)(27,52)(28,39)(57,110)(58,97)(59,112)(60,99)(61,86)(62,101)(63,88)(64,103)(65,90)(66,105)(67,92)(68,107)(69,94)(70,109)(71,96)(72,111)(73,98)(74,85)(75,100)(76,87)(77,102)(78,89)(79,104)(80,91)(81,106)(82,93)(83,108)(84,95), (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), (1,70,15,84)(2,83,16,69)(3,68,17,82)(4,81,18,67)(5,66,19,80)(6,79,20,65)(7,64,21,78)(8,77,22,63)(9,62,23,76)(10,75,24,61)(11,60,25,74)(12,73,26,59)(13,58,27,72)(14,71,28,57)(29,92,43,106)(30,105,44,91)(31,90,45,104)(32,103,46,89)(33,88,47,102)(34,101,48,87)(35,86,49,100)(36,99,50,85)(37,112,51,98)(38,97,52,111)(39,110,53,96)(40,95,54,109)(41,108,55,94)(42,93,56,107)>;
G:=Group( (1,89,15,103)(2,104,16,90)(3,91,17,105)(4,106,18,92)(5,93,19,107)(6,108,20,94)(7,95,21,109)(8,110,22,96)(9,97,23,111)(10,112,24,98)(11,99,25,85)(12,86,26,100)(13,101,27,87)(14,88,28,102)(29,81,43,67)(30,68,44,82)(31,83,45,69)(32,70,46,84)(33,57,47,71)(34,72,48,58)(35,59,49,73)(36,74,50,60)(37,61,51,75)(38,76,52,62)(39,63,53,77)(40,78,54,64)(41,65,55,79)(42,80,56,66), (1,54)(2,41)(3,56)(4,43)(5,30)(6,45)(7,32)(8,47)(9,34)(10,49)(11,36)(12,51)(13,38)(14,53)(15,40)(16,55)(17,42)(18,29)(19,44)(20,31)(21,46)(22,33)(23,48)(24,35)(25,50)(26,37)(27,52)(28,39)(57,110)(58,97)(59,112)(60,99)(61,86)(62,101)(63,88)(64,103)(65,90)(66,105)(67,92)(68,107)(69,94)(70,109)(71,96)(72,111)(73,98)(74,85)(75,100)(76,87)(77,102)(78,89)(79,104)(80,91)(81,106)(82,93)(83,108)(84,95), (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), (1,70,15,84)(2,83,16,69)(3,68,17,82)(4,81,18,67)(5,66,19,80)(6,79,20,65)(7,64,21,78)(8,77,22,63)(9,62,23,76)(10,75,24,61)(11,60,25,74)(12,73,26,59)(13,58,27,72)(14,71,28,57)(29,92,43,106)(30,105,44,91)(31,90,45,104)(32,103,46,89)(33,88,47,102)(34,101,48,87)(35,86,49,100)(36,99,50,85)(37,112,51,98)(38,97,52,111)(39,110,53,96)(40,95,54,109)(41,108,55,94)(42,93,56,107) );
G=PermutationGroup([[(1,89,15,103),(2,104,16,90),(3,91,17,105),(4,106,18,92),(5,93,19,107),(6,108,20,94),(7,95,21,109),(8,110,22,96),(9,97,23,111),(10,112,24,98),(11,99,25,85),(12,86,26,100),(13,101,27,87),(14,88,28,102),(29,81,43,67),(30,68,44,82),(31,83,45,69),(32,70,46,84),(33,57,47,71),(34,72,48,58),(35,59,49,73),(36,74,50,60),(37,61,51,75),(38,76,52,62),(39,63,53,77),(40,78,54,64),(41,65,55,79),(42,80,56,66)], [(1,54),(2,41),(3,56),(4,43),(5,30),(6,45),(7,32),(8,47),(9,34),(10,49),(11,36),(12,51),(13,38),(14,53),(15,40),(16,55),(17,42),(18,29),(19,44),(20,31),(21,46),(22,33),(23,48),(24,35),(25,50),(26,37),(27,52),(28,39),(57,110),(58,97),(59,112),(60,99),(61,86),(62,101),(63,88),(64,103),(65,90),(66,105),(67,92),(68,107),(69,94),(70,109),(71,96),(72,111),(73,98),(74,85),(75,100),(76,87),(77,102),(78,89),(79,104),(80,91),(81,106),(82,93),(83,108),(84,95)], [(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)], [(1,70,15,84),(2,83,16,69),(3,68,17,82),(4,81,18,67),(5,66,19,80),(6,79,20,65),(7,64,21,78),(8,77,22,63),(9,62,23,76),(10,75,24,61),(11,60,25,74),(12,73,26,59),(13,58,27,72),(14,71,28,57),(29,92,43,106),(30,105,44,91),(31,90,45,104),(32,103,46,89),(33,88,47,102),(34,101,48,87),(35,86,49,100),(36,99,50,85),(37,112,51,98),(38,97,52,111),(39,110,53,96),(40,95,54,109),(41,108,55,94),(42,93,56,107)]])
D4.10D14 is a maximal subgroup of
D4.9D28 D4.10D28 D28.38D4 D28.40D4 D4.11D28 D4.13D28 D8⋊11D14 D8.10D14 D8⋊6D14 D28.44D4 D28.33C23 D28.35C23 C14.C25 D14.C24 D7×2- 1+4
D4.10D14 is a maximal quotient of
C42.87D14 C42.89D14 C42.90D14 C42.91D14 C42.92D14 C42.94D14 C42.96D14 C42.98D14 C42.99D14 D4×Dic14 C42.105D14 C42.106D14 C42.108D14 D28⋊24D4 Dic14⋊23D4 D4⋊6D28 C42.115D14 C42.118D14 Dic14⋊10Q8 Q8⋊6Dic14 C42.125D14 Q8×D28 C42.134D14 C42.135D14 C14.682- 1+4 Dic14⋊19D4 C14.352+ 1+4 C14.712- 1+4 C14.722- 1+4 C14.732- 1+4 C14.492+ 1+4 C14.752- 1+4 C14.152- 1+4 C14.162- 1+4 Dic14⋊21D4 C14.772- 1+4 C14.572+ 1+4 C14.582+ 1+4 C14.792- 1+4 C14.802- 1+4 C14.602+ 1+4 C14.822- 1+4 C14.832- 1+4 C14.842- 1+4 C14.852- 1+4 C14.862- 1+4 C42.137D14 C42.139D14 C42.140D14 C42.141D14 Dic14⋊10D4 C42.144D14 C42.145D14 Dic14⋊7Q8 C42.147D14 C42.148D14 C42.152D14 C42.154D14 C42.155D14 C42.157D14 C42.159D14 C42.160D14 C42.161D14 C42.162D14 C42.164D14 C42.165D14 C14.1042- 1+4 C14.1052- 1+4 C14.1062- 1+4 C14.1072- 1+4 C14.1082- 1+4
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2- 1+4 | D4.10D14 |
| kernel | D4.10D14 | C2×Dic14 | C4○D28 | D4⋊2D7 | Q8×D7 | C7×C4○D4 | C4○D4 | C2×C4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 9 | 9 | 3 | 1 | 6 |
Matrix representation of D4.10D14 ►in GL4(𝔽29) generated by
| 6 | 0 | 28 | 28 |
| 0 | 6 | 2 | 3 |
| 24 | 8 | 23 | 0 |
| 13 | 21 | 0 | 23 |
| 4 | 19 | 28 | 1 |
| 21 | 25 | 0 | 24 |
| 8 | 19 | 15 | 10 |
| 0 | 19 | 19 | 14 |
| 11 | 8 | 19 | 3 |
| 18 | 0 | 7 | 7 |
| 7 | 5 | 26 | 21 |
| 11 | 16 | 8 | 21 |
| 16 | 18 | 18 | 11 |
| 10 | 13 | 26 | 0 |
| 0 | 20 | 18 | 27 |
| 13 | 20 | 2 | 11 |
G:=sub<GL(4,GF(29))| [6,0,24,13,0,6,8,21,28,2,23,0,28,3,0,23],[4,21,8,0,19,25,19,19,28,0,15,19,1,24,10,14],[11,18,7,11,8,0,5,16,19,7,26,8,3,7,21,21],[16,10,0,13,18,13,20,20,18,26,18,2,11,0,27,11] >;
D4.10D14 in GAP, Magma, Sage, TeX
D_4._{10}D_{14} % in TeX
G:=Group("D4.10D14"); // GroupNames label
G:=SmallGroup(224,186);
// by ID
G=gap.SmallGroup(224,186);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,86,579,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^14=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations