direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C14, C14.12C23, C28.20C22, C4.4(C2×C14), (C2×C28).9C2, (C2×C4).3C14, C22.4(C2×C14), C2.2(C22×C14), (C2×C14).15C22, SmallGroup(112,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C14
G = < a,b,c | a14=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C14
►Regular action on 112 points
Generators in S112
(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)]])
Q8×C14 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 | C7×Q8 |
| kernel | Q8×C14 | C2×C28 | C7×Q8 | C2×Q8 | C2×C4 | Q8 | C14 | C2 |
| # reps | 1 | 3 | 4 | 6 | 18 | 24 | 2 | 12 |
Matrix representation of Q8×C14 ►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] >;
Q8×C14 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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