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G = C7×C8.C4order 224 = 25·7

Direct product of C7 and C8.C4

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

Aliases: C7×C8.C4, C8.1C28, C56.5C4, C28.68D4, M4(2).2C14, (C2×C8).5C14, C4.8(C2×C28), C22.(C7×Q8), C4.19(C7×D4), (C2×C14).2Q8, (C2×C56).15C2, C28.45(C2×C4), C14.14(C4⋊C4), (C7×M4(2)).4C2, (C2×C28).119C22, C2.5(C7×C4⋊C4), (C2×C4).22(C2×C14), SmallGroup(224,57)

Series: Derived Chief Lower central Upper central

C1C4 — C7×C8.C4
C1C2C4C2×C4C2×C28C7×M4(2) — C7×C8.C4
C1C2C4 — C7×C8.C4
C1C28C2×C28 — C7×C8.C4

Generators and relations for C7×C8.C4
 G = < a,b,c | a7=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C14
2C8
2C8
2C56
2C56

Smallest permutation representation of C7×C8.C4
On 112 points
Generators in S112
(1 57 23 53 16 45 37)(2 58 24 54 9 46 38)(3 59 17 55 10 47 39)(4 60 18 56 11 48 40)(5 61 19 49 12 41 33)(6 62 20 50 13 42 34)(7 63 21 51 14 43 35)(8 64 22 52 15 44 36)(25 79 108 91 88 71 100)(26 80 109 92 81 72 101)(27 73 110 93 82 65 102)(28 74 111 94 83 66 103)(29 75 112 95 84 67 104)(30 76 105 96 85 68 97)(31 77 106 89 86 69 98)(32 78 107 90 87 70 99)
(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 97 3 103 5 101 7 99)(2 104 4 102 6 100 8 98)(9 95 11 93 13 91 15 89)(10 94 12 92 14 90 16 96)(17 74 19 80 21 78 23 76)(18 73 20 79 22 77 24 75)(25 64 31 58 29 60 27 62)(26 63 32 57 30 59 28 61)(33 72 35 70 37 68 39 66)(34 71 36 69 38 67 40 65)(41 81 43 87 45 85 47 83)(42 88 44 86 46 84 48 82)(49 109 51 107 53 105 55 111)(50 108 52 106 54 112 56 110)

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

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

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

C7×C8.C4 is a maximal subgroup of
C8.7Dic14  C8.Dic14  D56.C4  C56.8D4  Dic28.C4  M4(2).25D14  D5610C4  D567C4  C8.20D28  C8.21D28  C8.24D28

98 conjugacy classes

class 1 2A2B4A4B4C7A···7F8A8B8C8D8E8F8G8H14A···14F14G···14L28A···28L28M···28R56A···56X56Y···56AV
order1224447···78888888814···1414···1428···2828···2856···5656···56
size1121121···1222244441···12···21···12···22···24···4

98 irreducible representations

dim11111111222222
type++++-
imageC1C2C2C4C7C14C14C28D4Q8C8.C4C7×D4C7×Q8C7×C8.C4
kernelC7×C8.C4C2×C56C7×M4(2)C56C8.C4C2×C8M4(2)C8C28C2×C14C7C4C22C1
# reps11246612241146624

Matrix representation of C7×C8.C4 in GL2(𝔽113) generated by

280
028
,
440
018
,
01
150
G:=sub<GL(2,GF(113))| [28,0,0,28],[44,0,0,18],[0,15,1,0] >;

C7×C8.C4 in GAP, Magma, Sage, TeX

C_7\times C_8.C_4
% in TeX

G:=Group("C7xC8.C4");
// GroupNames label

G:=SmallGroup(224,57);
// by ID

G=gap.SmallGroup(224,57);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,175,3363,117,88]);
// Polycyclic

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

Export

Subgroup lattice of C7×C8.C4 in TeX

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