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## G = C7×M5(2)⋊C2order 448 = 26·7

### Direct product of C7 and M5(2)⋊C2

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×M5(2)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×C8.C4 — C7×M5(2)⋊C2
 Lower central C1 — C2 — C4 — C8 — C7×M5(2)⋊C2
 Upper central C1 — C14 — C2×C28 — C2×C56 — C7×M5(2)⋊C2

Generators and relations for C7×M5(2)⋊C2
G = < a,b,c,d | a7=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >

Subgroups: 162 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, C14, C14, C16, C2×C8, M4(2), D8, D8, C2×D4, C28, C2×C14, C2×C14, C8.C4, M5(2), C2×D8, C56, C56, C2×C28, C7×D4, C22×C14, M5(2)⋊C2, C112, C2×C56, C7×M4(2), C7×D8, C7×D8, D4×C14, C7×C8.C4, C7×M5(2), C14×D8, C7×M5(2)⋊C2
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, D4⋊C4, C2×C28, C7×D4, M5(2)⋊C2, C7×C22⋊C4, C7×D8, C7×SD16, C7×D4⋊C4, C7×M5(2)⋊C2

Smallest permutation representation of C7×M5(2)⋊C2
On 112 points
Generators in S112
(1 50 19 75 110 47 87)(2 51 20 76 111 48 88)(3 52 21 77 112 33 89)(4 53 22 78 97 34 90)(5 54 23 79 98 35 91)(6 55 24 80 99 36 92)(7 56 25 65 100 37 93)(8 57 26 66 101 38 94)(9 58 27 67 102 39 95)(10 59 28 68 103 40 96)(11 60 29 69 104 41 81)(12 61 30 70 105 42 82)(13 62 31 71 106 43 83)(14 63 32 72 107 44 84)(15 64 17 73 108 45 85)(16 49 18 74 109 46 86)
(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)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)
(2 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(17 21)(18 32)(20 30)(22 28)(23 31)(24 26)(25 29)(33 45)(34 40)(35 43)(36 38)(37 41)(42 48)(44 46)(49 63)(51 61)(52 64)(53 59)(54 62)(55 57)(56 60)(65 69)(66 80)(68 78)(70 76)(71 79)(72 74)(73 77)(81 93)(82 88)(83 91)(84 86)(85 89)(90 96)(92 94)(97 103)(98 106)(99 101)(100 104)(105 111)(107 109)(108 112)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 7A ··· 7F 8A 8B 8C 8D 8E 14A ··· 14F 14G ··· 14L 14M ··· 14X 16A 16B 16C 16D 28A ··· 28L 56A ··· 56L 56M ··· 56R 56S ··· 56AD 112A ··· 112X order 1 2 2 2 2 4 4 7 ··· 7 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 56 ··· 56 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 8 8 2 2 1 ··· 1 2 2 4 8 8 1 ··· 1 2 ··· 2 8 ··· 8 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D4 D4 D8 SD16 C7×D4 C7×D4 C7×D8 C7×SD16 M5(2)⋊C2 C7×M5(2)⋊C2 kernel C7×M5(2)⋊C2 C7×C8.C4 C7×M5(2) C14×D8 C7×D8 M5(2)⋊C2 C8.C4 M5(2) C2×D8 D8 C56 C2×C28 C28 C2×C14 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 4 6 6 6 6 24 1 1 2 2 6 6 12 12 2 12

Matrix representation of C7×M5(2)⋊C2 in GL4(𝔽113) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 13 15 111 0 100 98 1 112 100 16 51 49 95 97 62 64
,
 1 0 0 0 0 1 0 0 13 15 112 0 100 98 0 112
,
 1 0 0 0 112 112 0 0 48 36 31 82 1 64 82 82
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[13,100,100,95,15,98,16,97,111,1,51,62,0,112,49,64],[1,0,13,100,0,1,15,98,0,0,112,0,0,0,0,112],[1,112,48,1,0,112,36,64,0,0,31,82,0,0,82,82] >;

C7×M5(2)⋊C2 in GAP, Magma, Sage, TeX

C_7\times M_5(2)\rtimes C_2
% in TeX

G:=Group("C7xM5(2):C2");
// GroupNames label

G:=SmallGroup(448,165);
// by ID

G=gap.SmallGroup(448,165);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,5106,136,4911,172,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations

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