metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.10D14, C7⋊12- 1+4, C28.24C23, C14.10C24, D14.5C23, D28.13C22, Dic7.6C23, Dic14.13C22, (C2×Q8)⋊5D7, (Q8×D7)⋊4C2, C4○D28⋊6C2, (Q8×C14)⋊7C2, Q8⋊2D7⋊4C2, (C2×C4).24D14, (C4×D7).5C22, C4.24(C22×D7), C2.11(C23×D7), C7⋊D4.2C22, (C2×C14).68C23, (C2×C28).48C22, (C7×Q8).10C22, C22.7(C22×D7), SmallGroup(224,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.10D14
G = < a,b,c,d | a4=1, b2=c14=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c13 >
Subgroups: 510 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, Q8, D7, C14, C14, C2×Q8, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, 2- 1+4, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×Q8, C4○D28, Q8×D7, Q8⋊2D7, Q8×C14, Q8.10D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C22×D7, C23×D7, Q8.10D14
►On 112 points
(1 54 15 40)(2 55 16 41)(3 56 17 42)(4 29 18 43)(5 30 19 44)(6 31 20 45)(7 32 21 46)(8 33 22 47)(9 34 23 48)(10 35 24 49)(11 36 25 50)(12 37 26 51)(13 38 27 52)(14 39 28 53)(57 91 71 105)(58 92 72 106)(59 93 73 107)(60 94 74 108)(61 95 75 109)(62 96 76 110)(63 97 77 111)(64 98 78 112)(65 99 79 85)(66 100 80 86)(67 101 81 87)(68 102 82 88)(69 103 83 89)(70 104 84 90)
(1 84 15 70)(2 71 16 57)(3 58 17 72)(4 73 18 59)(5 60 19 74)(6 75 20 61)(7 62 21 76)(8 77 22 63)(9 64 23 78)(10 79 24 65)(11 66 25 80)(12 81 26 67)(13 68 27 82)(14 83 28 69)(29 93 43 107)(30 108 44 94)(31 95 45 109)(32 110 46 96)(33 97 47 111)(34 112 48 98)(35 99 49 85)(36 86 50 100)(37 101 51 87)(38 88 52 102)(39 103 53 89)(40 90 54 104)(41 105 55 91)(42 92 56 106)
(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 89 15 103)(2 102 16 88)(3 87 17 101)(4 100 18 86)(5 85 19 99)(6 98 20 112)(7 111 21 97)(8 96 22 110)(9 109 23 95)(10 94 24 108)(11 107 25 93)(12 92 26 106)(13 105 27 91)(14 90 28 104)(29 66 43 80)(30 79 44 65)(31 64 45 78)(32 77 46 63)(33 62 47 76)(34 75 48 61)(35 60 49 74)(36 73 50 59)(37 58 51 72)(38 71 52 57)(39 84 53 70)(40 69 54 83)(41 82 55 68)(42 67 56 81)
G:=sub<Sym(112)| (1,54,15,40)(2,55,16,41)(3,56,17,42)(4,29,18,43)(5,30,19,44)(6,31,20,45)(7,32,21,46)(8,33,22,47)(9,34,23,48)(10,35,24,49)(11,36,25,50)(12,37,26,51)(13,38,27,52)(14,39,28,53)(57,91,71,105)(58,92,72,106)(59,93,73,107)(60,94,74,108)(61,95,75,109)(62,96,76,110)(63,97,77,111)(64,98,78,112)(65,99,79,85)(66,100,80,86)(67,101,81,87)(68,102,82,88)(69,103,83,89)(70,104,84,90), (1,84,15,70)(2,71,16,57)(3,58,17,72)(4,73,18,59)(5,60,19,74)(6,75,20,61)(7,62,21,76)(8,77,22,63)(9,64,23,78)(10,79,24,65)(11,66,25,80)(12,81,26,67)(13,68,27,82)(14,83,28,69)(29,93,43,107)(30,108,44,94)(31,95,45,109)(32,110,46,96)(33,97,47,111)(34,112,48,98)(35,99,49,85)(36,86,50,100)(37,101,51,87)(38,88,52,102)(39,103,53,89)(40,90,54,104)(41,105,55,91)(42,92,56,106), (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), (1,89,15,103)(2,102,16,88)(3,87,17,101)(4,100,18,86)(5,85,19,99)(6,98,20,112)(7,111,21,97)(8,96,22,110)(9,109,23,95)(10,94,24,108)(11,107,25,93)(12,92,26,106)(13,105,27,91)(14,90,28,104)(29,66,43,80)(30,79,44,65)(31,64,45,78)(32,77,46,63)(33,62,47,76)(34,75,48,61)(35,60,49,74)(36,73,50,59)(37,58,51,72)(38,71,52,57)(39,84,53,70)(40,69,54,83)(41,82,55,68)(42,67,56,81)>;
G:=Group( (1,54,15,40)(2,55,16,41)(3,56,17,42)(4,29,18,43)(5,30,19,44)(6,31,20,45)(7,32,21,46)(8,33,22,47)(9,34,23,48)(10,35,24,49)(11,36,25,50)(12,37,26,51)(13,38,27,52)(14,39,28,53)(57,91,71,105)(58,92,72,106)(59,93,73,107)(60,94,74,108)(61,95,75,109)(62,96,76,110)(63,97,77,111)(64,98,78,112)(65,99,79,85)(66,100,80,86)(67,101,81,87)(68,102,82,88)(69,103,83,89)(70,104,84,90), (1,84,15,70)(2,71,16,57)(3,58,17,72)(4,73,18,59)(5,60,19,74)(6,75,20,61)(7,62,21,76)(8,77,22,63)(9,64,23,78)(10,79,24,65)(11,66,25,80)(12,81,26,67)(13,68,27,82)(14,83,28,69)(29,93,43,107)(30,108,44,94)(31,95,45,109)(32,110,46,96)(33,97,47,111)(34,112,48,98)(35,99,49,85)(36,86,50,100)(37,101,51,87)(38,88,52,102)(39,103,53,89)(40,90,54,104)(41,105,55,91)(42,92,56,106), (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), (1,89,15,103)(2,102,16,88)(3,87,17,101)(4,100,18,86)(5,85,19,99)(6,98,20,112)(7,111,21,97)(8,96,22,110)(9,109,23,95)(10,94,24,108)(11,107,25,93)(12,92,26,106)(13,105,27,91)(14,90,28,104)(29,66,43,80)(30,79,44,65)(31,64,45,78)(32,77,46,63)(33,62,47,76)(34,75,48,61)(35,60,49,74)(36,73,50,59)(37,58,51,72)(38,71,52,57)(39,84,53,70)(40,69,54,83)(41,82,55,68)(42,67,56,81) );
G=PermutationGroup([[(1,54,15,40),(2,55,16,41),(3,56,17,42),(4,29,18,43),(5,30,19,44),(6,31,20,45),(7,32,21,46),(8,33,22,47),(9,34,23,48),(10,35,24,49),(11,36,25,50),(12,37,26,51),(13,38,27,52),(14,39,28,53),(57,91,71,105),(58,92,72,106),(59,93,73,107),(60,94,74,108),(61,95,75,109),(62,96,76,110),(63,97,77,111),(64,98,78,112),(65,99,79,85),(66,100,80,86),(67,101,81,87),(68,102,82,88),(69,103,83,89),(70,104,84,90)], [(1,84,15,70),(2,71,16,57),(3,58,17,72),(4,73,18,59),(5,60,19,74),(6,75,20,61),(7,62,21,76),(8,77,22,63),(9,64,23,78),(10,79,24,65),(11,66,25,80),(12,81,26,67),(13,68,27,82),(14,83,28,69),(29,93,43,107),(30,108,44,94),(31,95,45,109),(32,110,46,96),(33,97,47,111),(34,112,48,98),(35,99,49,85),(36,86,50,100),(37,101,51,87),(38,88,52,102),(39,103,53,89),(40,90,54,104),(41,105,55,91),(42,92,56,106)], [(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)], [(1,89,15,103),(2,102,16,88),(3,87,17,101),(4,100,18,86),(5,85,19,99),(6,98,20,112),(7,111,21,97),(8,96,22,110),(9,109,23,95),(10,94,24,108),(11,107,25,93),(12,92,26,106),(13,105,27,91),(14,90,28,104),(29,66,43,80),(30,79,44,65),(31,64,45,78),(32,77,46,63),(33,62,47,76),(34,75,48,61),(35,60,49,74),(36,73,50,59),(37,58,51,72),(38,71,52,57),(39,84,53,70),(40,69,54,83),(41,82,55,68),(42,67,56,81)]])
Q8.10D14 is a maximal subgroup of
D28.4D4 D28.5D4 D28.14D4 D28.15D4 D28.29D4 D28.30D4 C56.C23 D28.44D4 C14.C25 D7×2- 1+4 D28.39C23
Q8.10D14 is a maximal quotient of
C14.72+ 1+4 C14.82+ 1+4 C14.2- 1+4 C14.2+ 1+4 C14.52- 1+4 C14.62- 1+4 C42.122D14 Q8⋊5Dic14 C42.125D14 C42.126D14 Q8⋊5D28 C42.132D14 C42.133D14 C42.134D14 C14.152- 1+4 C14.162- 1+4 C14.172- 1+4 D28⋊22D4 Dic14⋊22D4 C14.202- 1+4 C14.212- 1+4 C14.222- 1+4 C14.232- 1+4 C14.242- 1+4 C14.582+ 1+4 C14.262- 1+4 C42.147D14 C42.150D14 C42.151D14 C42.154D14 C42.157D14 C42.158D14 Dic14⋊8Q8 C42.171D14 D28⋊12D4 D28⋊8Q8 C42.174D14 C42.176D14 C42.177D14 C42.178D14 C42.180D14 C14.422- 1+4 Q8×C7⋊D4 C14.442- 1+4 C14.452- 1+4
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | 2- 1+4 | Q8.10D14 |
| kernel | Q8.10D14 | C4○D28 | Q8×D7 | Q8⋊2D7 | Q8×C14 | C2×Q8 | C2×C4 | Q8 | C7 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 3 | 9 | 12 | 1 | 6 |
Matrix representation of Q8.10D14 ►in GL4(𝔽29) generated by
| 0 | 0 | 4 | 15 |
| 0 | 0 | 28 | 25 |
| 25 | 14 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 2 | 22 | 8 | 0 |
| 14 | 27 | 0 | 8 |
| 8 | 0 | 27 | 7 |
| 0 | 8 | 15 | 2 |
| 0 | 0 | 9 | 10 |
| 0 | 0 | 9 | 23 |
| 20 | 19 | 0 | 0 |
| 20 | 6 | 0 | 0 |
| 18 | 18 | 23 | 7 |
| 9 | 11 | 23 | 6 |
| 6 | 22 | 18 | 18 |
| 6 | 23 | 9 | 11 |
G:=sub<GL(4,GF(29))| [0,0,25,1,0,0,14,4,4,28,0,0,15,25,0,0],[2,14,8,0,22,27,0,8,8,0,27,15,0,8,7,2],[0,0,20,20,0,0,19,6,9,9,0,0,10,23,0,0],[18,9,6,6,18,11,22,23,23,23,18,9,7,6,18,11] >;
Q8.10D14 in GAP, Magma, Sage, TeX
Q_8._{10}D_{14} % in TeX
G:=Group("Q8.10D14"); // GroupNames label
G:=SmallGroup(224,183);
// by ID
G=gap.SmallGroup(224,183);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,86,579,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^14=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations