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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C8.C4
 Lower central C1 — C2 — C4 — C7×C8.C4
 Upper central C1 — C28 — C2×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 >

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 2A 2B 4A 4B 4C 7A ··· 7F 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14F 14G ··· 14L 28A ··· 28L 28M ··· 28R 56A ··· 56X 56Y ··· 56AV order 1 2 2 4 4 4 7 ··· 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 1 1 2 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C7 C14 C14 C28 D4 Q8 C8.C4 C7×D4 C7×Q8 C7×C8.C4 kernel C7×C8.C4 C2×C56 C7×M4(2) C56 C8.C4 C2×C8 M4(2) C8 C28 C2×C14 C7 C4 C22 C1 # reps 1 1 2 4 6 6 12 24 1 1 4 6 6 24

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

 28 0 0 28
,
 44 0 0 18
,
 0 1 15 0
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

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