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G = C7×C8.C8order 448 = 26·7

Direct product of C7 and C8.C8

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

Aliases: C7×C8.C8, C8.1C56, C56.7C8, C56.22Q8, C56.109D4, C42.8C28, M5(2).4C14, C8.6(C7×Q8), C4.8(C2×C56), (C2×C8).9C28, C8.29(C7×D4), (C4×C8).10C14, (C4×C56).28C2, (C4×C28).23C4, C28.48(C2×C8), (C2×C56).28C4, C28.68(C4⋊C4), C14.15(C4⋊C8), (C7×M5(2)).8C2, (C2×C56).443C22, (C2×C14).17M4(2), C22.5(C7×M4(2)), C2.5(C7×C4⋊C8), C4.19(C7×C4⋊C4), (C2×C8).97(C2×C14), (C2×C4).68(C2×C28), (C2×C28).329(C2×C4), SmallGroup(448,168)

Series: Derived Chief Lower central Upper central

C1C4 — C7×C8.C8
C1C2C4C8C2×C8C2×C56C7×M5(2) — C7×C8.C8
C1C2C4 — C7×C8.C8
C1C56C2×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 >

2C2
2C4
2C4
2C14
2C2×C4
2C28
2C28
2C16
2C16
2C2×C28
2C112
2C112

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 2A2B4A4B4C···4G7A···7F8A8B8C8D8E···8J14A···14F14G···14L16A···16H28A···28L28M···28AP56A···56X56Y···56BH112A···112AV
order122444···47···788888···814···1414···1416···1628···2828···2856···5656···56112···112
size112112···21···111112···21···12···24···41···12···21···12···24···4

196 irreducible representations

dim11111111111122222222
type++++-
imageC1C2C2C4C4C7C8C14C14C28C28C56D4Q8M4(2)C7×D4C7×Q8C8.C8C7×M4(2)C7×C8.C8
kernelC7×C8.C8C4×C56C7×M5(2)C4×C28C2×C56C8.C8C56C4×C8M5(2)C42C2×C8C8C56C56C2×C14C8C8C7C22C1
# reps11222686121212481126681248

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

280
028
,
950
044
,
01
950
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

Subgroup lattice of C7×C8.C8 in TeX

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