Copied to
clipboard

## G = C7×C8.C8order 448 = 26·7

### Direct product of C7 and C8.C8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C8.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C56 — C7×M5(2) — C7×C8.C8
 Lower central C1 — C2 — C4 — C7×C8.C8
 Upper central C1 — C56 — C2×C56 — C7×C8.C8

Generators and relations for C7×C8.C8
G = < a,b,c | a7=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >

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

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

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

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

196 conjugacy classes

 class 1 2A 2B 4A 4B 4C ··· 4G 7A ··· 7F 8A 8B 8C 8D 8E ··· 8J 14A ··· 14F 14G ··· 14L 16A ··· 16H 28A ··· 28L 28M ··· 28AP 56A ··· 56X 56Y ··· 56BH 112A ··· 112AV order 1 2 2 4 4 4 ··· 4 7 ··· 7 8 8 8 8 8 ··· 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 1 1 2 ··· 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C4 C7 C8 C14 C14 C28 C28 C56 D4 Q8 M4(2) C7×D4 C7×Q8 C8.C8 C7×M4(2) C7×C8.C8 kernel C7×C8.C8 C4×C56 C7×M5(2) C4×C28 C2×C56 C8.C8 C56 C4×C8 M5(2) C42 C2×C8 C8 C56 C56 C2×C14 C8 C8 C7 C22 C1 # reps 1 1 2 2 2 6 8 6 12 12 12 48 1 1 2 6 6 8 12 48

Matrix representation of C7×C8.C8 in GL2(𝔽113) generated by

 28 0 0 28
,
 95 0 0 44
,
 0 1 95 0
G:=sub<GL(2,GF(113))| [28,0,0,28],[95,0,0,44],[0,95,1,0] >;

C7×C8.C8 in GAP, Magma, Sage, TeX

C_7\times C_8.C_8
% in TeX

G:=Group("C7xC8.C8");
// GroupNames label

G:=SmallGroup(448,168);
// by ID

G=gap.SmallGroup(448,168);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,7059,248,102,124]);
// Polycyclic

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

Export

׿
×
𝔽