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## G = Q8×C14order 112 = 24·7

### Direct product of C14 and Q8

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

 Derived series C1 — C2 — Q8×C14
 Chief series C1 — C2 — C14 — C28 — C7×Q8 — Q8×C14
 Lower central C1 — C2 — Q8×C14
 Upper central C1 — C2×C14 — Q8×C14

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  D143Q8  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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