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## G = C7×M5(2)  order 224 = 25·7

### Direct product of C7 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×M5(2)
 Chief series C1 — C2 — C4 — C8 — C56 — C112 — C7×M5(2)
 Lower central C1 — C2 — C7×M5(2)
 Upper central C1 — C56 — C7×M5(2)

Generators and relations for C7×M5(2)
G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C7×M5(2)
On 112 points
Generators in S112
(1 105 42 21 71 82 50)(2 106 43 22 72 83 51)(3 107 44 23 73 84 52)(4 108 45 24 74 85 53)(5 109 46 25 75 86 54)(6 110 47 26 76 87 55)(7 111 48 27 77 88 56)(8 112 33 28 78 89 57)(9 97 34 29 79 90 58)(10 98 35 30 80 91 59)(11 99 36 31 65 92 60)(12 100 37 32 66 93 61)(13 101 38 17 67 94 62)(14 102 39 18 68 95 63)(15 103 40 19 69 96 64)(16 104 41 20 70 81 49)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)

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

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

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

C7×M5(2) is a maximal subgroup of
C56.9Q8  C112⋊C4  C16⋊Dic7  M5(2)⋊D7  Dic14.C8  C28.3D8  C28.4D8  D562C4  C16.12D14  C16⋊D14  C16.D14

140 conjugacy classes

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

140 irreducible representations

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

Matrix representation of C7×M5(2) in GL3(𝔽113) generated by

 28 0 0 0 1 0 0 0 1
,
 112 0 0 0 3 111 0 52 110
,
 112 0 0 0 1 0 0 3 112
G:=sub<GL(3,GF(113))| [28,0,0,0,1,0,0,0,1],[112,0,0,0,3,52,0,111,110],[112,0,0,0,1,3,0,0,112] >;

C7×M5(2) in GAP, Magma, Sage, TeX

C_7\times M_5(2)
% in TeX

G:=Group("C7xM5(2)");
// GroupNames label

G:=SmallGroup(224,59);
// by ID

G=gap.SmallGroup(224,59);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,168,1369,69,88]);
// Polycyclic

G:=Group<a,b,c|a^7=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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