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## G = D7×D16order 448 = 26·7

### Direct product of D7 and D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D7×D16
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×D8 — D7×D16
 Lower central C7 — C14 — C28 — C56 — D7×D16
 Upper central C1 — C2 — C4 — C8 — D16

Generators and relations for D7×D16
G = < a,b,c,d | a7=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 880 in 98 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, C23, D7, D7, C14, C14, C16, C16, C2×C8, D8, D8, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C16, D16, D16, C2×D8, C7⋊C8, C56, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C2×D16, C7⋊C16, C112, C8×D7, D56, D4⋊D7, C7×D8, D4×D7, D7×C16, D112, C7⋊D16, C7×D16, D7×D8, D7×D16
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, D16, C2×D8, C22×D7, C2×D16, D4×D7, D7×D8, D7×D16

Smallest permutation representation of D7×D16
On 112 points
Generators in S112
(1 98 61 70 24 89 34)(2 99 62 71 25 90 35)(3 100 63 72 26 91 36)(4 101 64 73 27 92 37)(5 102 49 74 28 93 38)(6 103 50 75 29 94 39)(7 104 51 76 30 95 40)(8 105 52 77 31 96 41)(9 106 53 78 32 81 42)(10 107 54 79 17 82 43)(11 108 55 80 18 83 44)(12 109 56 65 19 84 45)(13 110 57 66 20 85 46)(14 111 58 67 21 86 47)(15 112 59 68 22 87 48)(16 97 60 69 23 88 33)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 33)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 97)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 66)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)(97 106)(98 105)(99 104)(100 103)(101 102)(107 112)(108 111)(109 110)

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D16 D4×D7 D7×D8 D7×D16 kernel D7×D16 D7×C16 D112 C7⋊D16 C7×D16 D7×D8 C7⋊C8 C4×D7 D16 Dic7 D14 C16 D8 D7 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 3 2 2 3 6 8 3 6 12

Matrix representation of D7×D16 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
,
 112 0 0 0 0 112 0 0 0 0 14 57 0 0 60 91
,
 112 0 0 0 0 112 0 0 0 0 99 56 0 0 51 14
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],[112,0,0,0,0,112,0,0,0,0,14,60,0,0,57,91],[112,0,0,0,0,112,0,0,0,0,99,51,0,0,56,14] >;

D7×D16 in GAP, Magma, Sage, TeX

D_7\times D_{16}
% in TeX

G:=Group("D7xD16");
// GroupNames label

G:=SmallGroup(448,444);
// by ID

G=gap.SmallGroup(448,444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,135,346,185,192,851,438,102,18822]);
// 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^-1>;
// generators/relations

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