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G = Q8xC14order 112 = 24·7

Direct product of C14 and Q8

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

C1C2 — Q8xC14
C1C2C14C28C7xQ8 — Q8xC14
C1C2 — Q8xC14
C1C2xC14 — Q8xC14

Generators and relations for Q8xC14
 G = < a,b,c | a14=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 38, all normal (8 characteristic)
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2xQ8, C2xC14, C7xQ8, C22xC14, Q8xC14

Smallest permutation representation of Q8xC14
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)]])

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 2A2B2C4A···4F7A···7F14A···14R28A···28AJ
order12224···47···714···1428···28
size11112···21···11···12···2

70 irreducible representations

dim11111122
type+++-
imageC1C2C2C7C14C14Q8C7xQ8
kernelQ8xC14C2xC28C7xQ8C2xQ8C2xC4Q8C14C2
# reps13461824212

Matrix representation of Q8xC14 in GL4(F29) generated by

7000
02800
0010
0001
,
1000
0100
00028
0010
,
28000
02800
00158
00814
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

Export

Subgroup lattice of Q8xC14 in TeX

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