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## G = C7×C8.Q8order 448 = 26·7

### Direct product of C7 and C8.Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×C8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×C4.Q8 — C7×C8.Q8
 Lower central C1 — C2 — C4 — C8 — C7×C8.Q8
 Upper central C1 — C14 — C2×C28 — C2×C56 — C7×C8.Q8

Generators and relations for C7×C8.Q8
G = < a,b,c,d | a7=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >

Smallest permutation representation of C7×C8.Q8
On 112 points
Generators in S112
(1 39 31 58 73 87 106)(2 40 32 59 74 88 107)(3 41 17 60 75 89 108)(4 42 18 61 76 90 109)(5 43 19 62 77 91 110)(6 44 20 63 78 92 111)(7 45 21 64 79 93 112)(8 46 22 49 80 94 97)(9 47 23 50 65 95 98)(10 48 24 51 66 96 99)(11 33 25 52 67 81 100)(12 34 26 53 68 82 101)(13 35 27 54 69 83 102)(14 36 28 55 70 84 103)(15 37 29 56 71 85 104)(16 38 30 57 72 86 105)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 28 22 32 26 20 30 24)(33 35 37 39 41 43 45 47)(34 44 38 48 42 36 46 40)(49 59 53 63 57 51 61 55)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 76 70 80 74 68 78 72)(81 83 85 87 89 91 93 95)(82 92 86 96 90 84 94 88)(97 107 101 111 105 99 109 103)(98 100 102 104 106 108 110 112)
(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)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 21)(18 24 26 32)(19 27)(20 30 28 22)(25 29)(33 37)(34 40 42 48)(35 43)(36 46 44 38)(41 45)(49 63 57 55)(51 53 59 61)(52 56)(54 62)(60 64)(66 68 74 76)(67 71)(69 77)(70 80 78 72)(75 79)(81 85)(82 88 90 96)(83 91)(84 94 92 86)(89 93)(97 111 105 103)(99 101 107 109)(100 104)(102 110)(108 112)

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

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

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

112 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 8E 14A ··· 14F 14G ··· 14L 16A 16B 16C 16D 28A ··· 28L 28M ··· 28X 56A ··· 56L 56M ··· 56R 56S ··· 56AD 112A ··· 112X order 1 2 2 4 4 4 4 7 ··· 7 8 8 8 8 8 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 2 2 8 8 1 ··· 1 2 2 4 8 8 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 Q8 D4 SD16 SD16 C7×Q8 C7×D4 C7×SD16 C7×SD16 C8.Q8 C7×C8.Q8 kernel C7×C8.Q8 C7×C4.Q8 C7×C8.C4 C7×M5(2) C112 C8.Q8 C4.Q8 C8.C4 M5(2) C16 C56 C2×C28 C28 C2×C14 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 4 6 6 6 6 24 1 1 2 2 6 6 12 12 2 12

Matrix representation of C7×C8.Q8 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 109 0 0 0 0 109
,
 13 100 0 0 13 13 0 0 0 0 100 13 0 0 100 100
,
 0 0 1 0 0 0 0 1 13 100 0 0 13 13 0 0
,
 1 0 0 0 0 112 0 0 0 0 100 13 0 0 13 13
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[13,13,0,0,100,13,0,0,0,0,100,100,0,0,13,100],[0,0,13,13,0,0,100,13,1,0,0,0,0,1,0,0],[1,0,0,0,0,112,0,0,0,0,100,13,0,0,13,13] >;

C7×C8.Q8 in GAP, Magma, Sage, TeX

C_7\times C_8.Q_8
% in TeX

G:=Group("C7xC8.Q8");
// GroupNames label

G:=SmallGroup(448,169);
// by ID

G=gap.SmallGroup(448,169);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,3923,136,4911,14117,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations

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