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## G = D7×M5(2)  order 448 = 26·7

### Direct product of D7 and M5(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×M5(2)
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×C2×C8 — D7×M5(2)
 Lower central C7 — C14 — D7×M5(2)
 Upper central C1 — C8 — M5(2)

Generators and relations for D7×M5(2)
G = < a,b,c,d | a7=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 308 in 90 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, D7, C14, C14, C16, C16, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, D14, C2×C14, C2×C16, M5(2), M5(2), C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C2×M5(2), C7⋊C16, C112, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, D7×C16, C16⋊D7, C28.C8, C7×M5(2), D7×C2×C8, D7×M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, M5(2), C22×C8, C4×D7, C22×D7, C2×M5(2), C8×D7, C2×C4×D7, D7×C2×C8, D7×M5(2)

Smallest permutation representation of D7×M5(2)
On 112 points
Generators in S112
(1 47 29 59 79 88 101)(2 48 30 60 80 89 102)(3 33 31 61 65 90 103)(4 34 32 62 66 91 104)(5 35 17 63 67 92 105)(6 36 18 64 68 93 106)(7 37 19 49 69 94 107)(8 38 20 50 70 95 108)(9 39 21 51 71 96 109)(10 40 22 52 72 81 110)(11 41 23 53 73 82 111)(12 42 24 54 74 83 112)(13 43 25 55 75 84 97)(14 44 26 56 76 85 98)(15 45 27 57 77 86 99)(16 46 28 58 78 87 100)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 97)(14 98)(15 99)(16 100)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 65)(32 66)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)
(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)(50 58)(52 60)(54 62)(56 64)(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)| (1,47,29,59,79,88,101)(2,48,30,60,80,89,102)(3,33,31,61,65,90,103)(4,34,32,62,66,91,104)(5,35,17,63,67,92,105)(6,36,18,64,68,93,106)(7,37,19,49,69,94,107)(8,38,20,50,70,95,108)(9,39,21,51,71,96,109)(10,40,22,52,72,81,110)(11,41,23,53,73,82,111)(12,42,24,54,74,83,112)(13,43,25,55,75,84,97)(14,44,26,56,76,85,98)(15,45,27,57,77,86,99)(16,46,28,58,78,87,100), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,97)(14,98)(15,99)(16,100)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,65)(32,66)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89), (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)(50,58)(52,60)(54,62)(56,64)(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( (1,47,29,59,79,88,101)(2,48,30,60,80,89,102)(3,33,31,61,65,90,103)(4,34,32,62,66,91,104)(5,35,17,63,67,92,105)(6,36,18,64,68,93,106)(7,37,19,49,69,94,107)(8,38,20,50,70,95,108)(9,39,21,51,71,96,109)(10,40,22,52,72,81,110)(11,41,23,53,73,82,111)(12,42,24,54,74,83,112)(13,43,25,55,75,84,97)(14,44,26,56,76,85,98)(15,45,27,57,77,86,99)(16,46,28,58,78,87,100), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,97)(14,98)(15,99)(16,100)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,65)(32,66)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89), (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)(50,58)(52,60)(54,62)(56,64)(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([[(1,47,29,59,79,88,101),(2,48,30,60,80,89,102),(3,33,31,61,65,90,103),(4,34,32,62,66,91,104),(5,35,17,63,67,92,105),(6,36,18,64,68,93,106),(7,37,19,49,69,94,107),(8,38,20,50,70,95,108),(9,39,21,51,71,96,109),(10,40,22,52,72,81,110),(11,41,23,53,73,82,111),(12,42,24,54,74,83,112),(13,43,25,55,75,84,97),(14,44,26,56,76,85,98),(15,45,27,57,77,86,99),(16,46,28,58,78,87,100)], [(1,101),(2,102),(3,103),(4,104),(5,105),(6,106),(7,107),(8,108),(9,109),(10,110),(11,111),(12,112),(13,97),(14,98),(15,99),(16,100),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,65),(32,66),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89)], [(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),(50,58),(52,60),(54,62),(56,64),(66,74),(68,76),(70,78),(72,80),(81,89),(83,91),(85,93),(87,95),(98,106),(100,108),(102,110),(104,112)]])

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 14A 14B 14C 14D 14E 14F 16A ··· 16H 16I ··· 16P 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 14 14 14 14 14 14 16 ··· 16 16 ··· 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 7 7 14 1 1 2 7 7 14 2 2 2 1 1 1 1 2 2 7 7 7 7 14 14 2 2 2 4 4 4 2 ··· 2 14 ··· 14 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D7 D14 D14 M5(2) C4×D7 C4×D7 C8×D7 C8×D7 D7×M5(2) kernel D7×M5(2) D7×C16 C16⋊D7 C28.C8 C7×M5(2) D7×C2×C8 C8×D7 C2×C7⋊C8 C2×C4×D7 C4×D7 C2×Dic7 C22×D7 M5(2) C16 C2×C8 D7 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 8 4 4 3 6 3 8 6 6 12 12 12

Matrix representation of D7×M5(2) in GL4(𝔽113) generated by

 0 1 0 0 112 79 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 44 0 0 0 0 44 0 0 0 0 80 81 0 0 24 33
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 5 112
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[44,0,0,0,0,44,0,0,0,0,80,24,0,0,81,33],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,0,112] >;

D7×M5(2) in GAP, Magma, Sage, TeX

D_7\times M_5(2)
% in TeX

G:=Group("D7xM5(2)");
// GroupNames label

G:=SmallGroup(448,440);
// by ID

G=gap.SmallGroup(448,440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,58,80,102,18822]);
// Polycyclic

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

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