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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 70 48 85)(2 57 49 86)(3 58 50 87)(4 59 51 88)(5 60 52 89)(6 61 53 90)(7 62 54 91)(8 63 55 92)(9 64 56 93)(10 65 43 94)(11 66 44 95)(12 67 45 96)(13 68 46 97)(14 69 47 98)(15 107 76 37)(16 108 77 38)(17 109 78 39)(18 110 79 40)(19 111 80 41)(20 112 81 42)(21 99 82 29)(22 100 83 30)(23 101 84 31)(24 102 71 32)(25 103 72 33)(26 104 73 34)(27 105 74 35)(28 106 75 36)
(1 41 48 111)(2 42 49 112)(3 29 50 99)(4 30 51 100)(5 31 52 101)(6 32 53 102)(7 33 54 103)(8 34 55 104)(9 35 56 105)(10 36 43 106)(11 37 44 107)(12 38 45 108)(13 39 46 109)(14 40 47 110)(15 66 76 95)(16 67 77 96)(17 68 78 97)(18 69 79 98)(19 70 80 85)(20 57 81 86)(21 58 82 87)(22 59 83 88)(23 60 84 89)(24 61 71 90)(25 62 72 91)(26 63 73 92)(27 64 74 93)(28 65 75 94)

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), (1,70,48,85)(2,57,49,86)(3,58,50,87)(4,59,51,88)(5,60,52,89)(6,61,53,90)(7,62,54,91)(8,63,55,92)(9,64,56,93)(10,65,43,94)(11,66,44,95)(12,67,45,96)(13,68,46,97)(14,69,47,98)(15,107,76,37)(16,108,77,38)(17,109,78,39)(18,110,79,40)(19,111,80,41)(20,112,81,42)(21,99,82,29)(22,100,83,30)(23,101,84,31)(24,102,71,32)(25,103,72,33)(26,104,73,34)(27,105,74,35)(28,106,75,36), (1,41,48,111)(2,42,49,112)(3,29,50,99)(4,30,51,100)(5,31,52,101)(6,32,53,102)(7,33,54,103)(8,34,55,104)(9,35,56,105)(10,36,43,106)(11,37,44,107)(12,38,45,108)(13,39,46,109)(14,40,47,110)(15,66,76,95)(16,67,77,96)(17,68,78,97)(18,69,79,98)(19,70,80,85)(20,57,81,86)(21,58,82,87)(22,59,83,88)(23,60,84,89)(24,61,71,90)(25,62,72,91)(26,63,73,92)(27,64,74,93)(28,65,75,94)>;

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), (1,70,48,85)(2,57,49,86)(3,58,50,87)(4,59,51,88)(5,60,52,89)(6,61,53,90)(7,62,54,91)(8,63,55,92)(9,64,56,93)(10,65,43,94)(11,66,44,95)(12,67,45,96)(13,68,46,97)(14,69,47,98)(15,107,76,37)(16,108,77,38)(17,109,78,39)(18,110,79,40)(19,111,80,41)(20,112,81,42)(21,99,82,29)(22,100,83,30)(23,101,84,31)(24,102,71,32)(25,103,72,33)(26,104,73,34)(27,105,74,35)(28,106,75,36), (1,41,48,111)(2,42,49,112)(3,29,50,99)(4,30,51,100)(5,31,52,101)(6,32,53,102)(7,33,54,103)(8,34,55,104)(9,35,56,105)(10,36,43,106)(11,37,44,107)(12,38,45,108)(13,39,46,109)(14,40,47,110)(15,66,76,95)(16,67,77,96)(17,68,78,97)(18,69,79,98)(19,70,80,85)(20,57,81,86)(21,58,82,87)(22,59,83,88)(23,60,84,89)(24,61,71,90)(25,62,72,91)(26,63,73,92)(27,64,74,93)(28,65,75,94) );

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)], [(1,70,48,85),(2,57,49,86),(3,58,50,87),(4,59,51,88),(5,60,52,89),(6,61,53,90),(7,62,54,91),(8,63,55,92),(9,64,56,93),(10,65,43,94),(11,66,44,95),(12,67,45,96),(13,68,46,97),(14,69,47,98),(15,107,76,37),(16,108,77,38),(17,109,78,39),(18,110,79,40),(19,111,80,41),(20,112,81,42),(21,99,82,29),(22,100,83,30),(23,101,84,31),(24,102,71,32),(25,103,72,33),(26,104,73,34),(27,105,74,35),(28,106,75,36)], [(1,41,48,111),(2,42,49,112),(3,29,50,99),(4,30,51,100),(5,31,52,101),(6,32,53,102),(7,33,54,103),(8,34,55,104),(9,35,56,105),(10,36,43,106),(11,37,44,107),(12,38,45,108),(13,39,46,109),(14,40,47,110),(15,66,76,95),(16,67,77,96),(17,68,78,97),(18,69,79,98),(19,70,80,85),(20,57,81,86),(21,58,82,87),(22,59,83,88),(23,60,84,89),(24,61,71,90),(25,62,72,91),(26,63,73,92),(27,64,74,93),(28,65,75,94)])

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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