direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8xC14, C14.12C23, C28.20C22, C4.4(C2xC14), (C2xC28).9C2, (C2xC4).3C14, C22.4(C2xC14), C2.2(C22xC14), (C2xC14).15C22, SmallGroup(112,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8xC14
G = < a,b,c | a14=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
(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 45 98 19)(2 46 85 20)(3 47 86 21)(4 48 87 22)(5 49 88 23)(6 50 89 24)(7 51 90 25)(8 52 91 26)(9 53 92 27)(10 54 93 28)(11 55 94 15)(12 56 95 16)(13 43 96 17)(14 44 97 18)(29 60 101 80)(30 61 102 81)(31 62 103 82)(32 63 104 83)(33 64 105 84)(34 65 106 71)(35 66 107 72)(36 67 108 73)(37 68 109 74)(38 69 110 75)(39 70 111 76)(40 57 112 77)(41 58 99 78)(42 59 100 79)
(1 105 98 33)(2 106 85 34)(3 107 86 35)(4 108 87 36)(5 109 88 37)(6 110 89 38)(7 111 90 39)(8 112 91 40)(9 99 92 41)(10 100 93 42)(11 101 94 29)(12 102 95 30)(13 103 96 31)(14 104 97 32)(15 80 55 60)(16 81 56 61)(17 82 43 62)(18 83 44 63)(19 84 45 64)(20 71 46 65)(21 72 47 66)(22 73 48 67)(23 74 49 68)(24 75 50 69)(25 76 51 70)(26 77 52 57)(27 78 53 58)(28 79 54 59)
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), (1,45,98,19)(2,46,85,20)(3,47,86,21)(4,48,87,22)(5,49,88,23)(6,50,89,24)(7,51,90,25)(8,52,91,26)(9,53,92,27)(10,54,93,28)(11,55,94,15)(12,56,95,16)(13,43,96,17)(14,44,97,18)(29,60,101,80)(30,61,102,81)(31,62,103,82)(32,63,104,83)(33,64,105,84)(34,65,106,71)(35,66,107,72)(36,67,108,73)(37,68,109,74)(38,69,110,75)(39,70,111,76)(40,57,112,77)(41,58,99,78)(42,59,100,79), (1,105,98,33)(2,106,85,34)(3,107,86,35)(4,108,87,36)(5,109,88,37)(6,110,89,38)(7,111,90,39)(8,112,91,40)(9,99,92,41)(10,100,93,42)(11,101,94,29)(12,102,95,30)(13,103,96,31)(14,104,97,32)(15,80,55,60)(16,81,56,61)(17,82,43,62)(18,83,44,63)(19,84,45,64)(20,71,46,65)(21,72,47,66)(22,73,48,67)(23,74,49,68)(24,75,50,69)(25,76,51,70)(26,77,52,57)(27,78,53,58)(28,79,54,59)>;
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), (1,45,98,19)(2,46,85,20)(3,47,86,21)(4,48,87,22)(5,49,88,23)(6,50,89,24)(7,51,90,25)(8,52,91,26)(9,53,92,27)(10,54,93,28)(11,55,94,15)(12,56,95,16)(13,43,96,17)(14,44,97,18)(29,60,101,80)(30,61,102,81)(31,62,103,82)(32,63,104,83)(33,64,105,84)(34,65,106,71)(35,66,107,72)(36,67,108,73)(37,68,109,74)(38,69,110,75)(39,70,111,76)(40,57,112,77)(41,58,99,78)(42,59,100,79), (1,105,98,33)(2,106,85,34)(3,107,86,35)(4,108,87,36)(5,109,88,37)(6,110,89,38)(7,111,90,39)(8,112,91,40)(9,99,92,41)(10,100,93,42)(11,101,94,29)(12,102,95,30)(13,103,96,31)(14,104,97,32)(15,80,55,60)(16,81,56,61)(17,82,43,62)(18,83,44,63)(19,84,45,64)(20,71,46,65)(21,72,47,66)(22,73,48,67)(23,74,49,68)(24,75,50,69)(25,76,51,70)(26,77,52,57)(27,78,53,58)(28,79,54,59) );
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)], [(1,45,98,19),(2,46,85,20),(3,47,86,21),(4,48,87,22),(5,49,88,23),(6,50,89,24),(7,51,90,25),(8,52,91,26),(9,53,92,27),(10,54,93,28),(11,55,94,15),(12,56,95,16),(13,43,96,17),(14,44,97,18),(29,60,101,80),(30,61,102,81),(31,62,103,82),(32,63,104,83),(33,64,105,84),(34,65,106,71),(35,66,107,72),(36,67,108,73),(37,68,109,74),(38,69,110,75),(39,70,111,76),(40,57,112,77),(41,58,99,78),(42,59,100,79)], [(1,105,98,33),(2,106,85,34),(3,107,86,35),(4,108,87,36),(5,109,88,37),(6,110,89,38),(7,111,90,39),(8,112,91,40),(9,99,92,41),(10,100,93,42),(11,101,94,29),(12,102,95,30),(13,103,96,31),(14,104,97,32),(15,80,55,60),(16,81,56,61),(17,82,43,62),(18,83,44,63),(19,84,45,64),(20,71,46,65),(21,72,47,66),(22,73,48,67),(23,74,49,68),(24,75,50,69),(25,76,51,70),(26,77,52,57),(27,78,53,58),(28,79,54,59)]])
Q8xC14 is a maximal subgroup of
Q8:Dic7 C28.10D4 C28.C23 Dic7:Q8 D14:3Q8 C28.23D4 Q8.10D14
70 conjugacy classes
class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28AJ |
order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
70 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
type | + | + | + | - | ||||
image | C1 | C2 | C2 | C7 | C14 | C14 | Q8 | C7xQ8 |
kernel | Q8xC14 | C2xC28 | C7xQ8 | C2xQ8 | C2xC4 | Q8 | C14 | C2 |
# reps | 1 | 3 | 4 | 6 | 18 | 24 | 2 | 12 |
Matrix representation of Q8xC14 ►in GL4(F29) generated by
7 | 0 | 0 | 0 |
0 | 28 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 0 | 28 |
0 | 0 | 1 | 0 |
28 | 0 | 0 | 0 |
0 | 28 | 0 | 0 |
0 | 0 | 15 | 8 |
0 | 0 | 8 | 14 |
G:=sub<GL(4,GF(29))| [7,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,28,0],[28,0,0,0,0,28,0,0,0,0,15,8,0,0,8,14] >;
Q8xC14 in GAP, Magma, Sage, TeX
Q_8\times C_{14}
% in TeX
G:=Group("Q8xC14");
// GroupNames label
G:=SmallGroup(112,39);
// by ID
G=gap.SmallGroup(112,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-2,280,581,286]);
// Polycyclic
G:=Group<a,b,c|a^14=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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