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## G = D7×Q16order 224 = 25·7

### Direct product of D7 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×Q16
 Chief series C1 — C7 — C14 — C28 — C4×D7 — Q8×D7 — D7×Q16
 Lower central C7 — C14 — C28 — D7×Q16
 Upper central C1 — C2 — C4 — Q16

Generators and relations for D7×Q16
G = < a,b,c,d | a7=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 238 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, Q8, Q8, D7, C14, C2×C8, Q16, Q16, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C7×Q8, C8×D7, Dic28, C7⋊Q16, C7×Q16, Q8×D7, D7×Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C2×Q16, C22×D7, D4×D7, D7×Q16

Smallest permutation representation of D7×Q16
On 112 points
Generators in S112
(1 102 21 12 86 40 50)(2 103 22 13 87 33 51)(3 104 23 14 88 34 52)(4 97 24 15 81 35 53)(5 98 17 16 82 36 54)(6 99 18 9 83 37 55)(7 100 19 10 84 38 56)(8 101 20 11 85 39 49)(25 74 43 59 70 91 109)(26 75 44 60 71 92 110)(27 76 45 61 72 93 111)(28 77 46 62 65 94 112)(29 78 47 63 66 95 105)(30 79 48 64 67 96 106)(31 80 41 57 68 89 107)(32 73 42 58 69 90 108)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 13)(10 14)(11 15)(12 16)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 97)(40 98)(41 72)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(57 61)(58 62)(59 63)(60 64)(73 94)(74 95)(75 96)(76 89)(77 90)(78 91)(79 92)(80 93)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 62 13 58)(10 61 14 57)(11 60 15 64)(12 59 16 63)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(33 90 37 94)(34 89 38 93)(35 96 39 92)(36 95 40 91)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 87 69 83)(66 86 70 82)(67 85 71 81)(68 84 72 88)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)

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

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

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

D7×Q16 is a maximal subgroup of
SD32⋊D7  Q32⋊D7  D28.30D4  D8.10D14  D28.44D4
D7×Q16 is a maximal quotient of
Dic74Q16  Dic7.1Q16  Dic7⋊Q16  Dic7.Q16  D144Q16  D14.Q16  D14⋊Q16  Dic286C4  C562Q8  Dic142Q8  D14.2Q16  D142Q16  C56.26D4  Dic73Q16  D145Q16  D143Q16

35 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 28A 28B 28C 28D ··· 28I 56A ··· 56F order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 28 28 28 28 ··· 28 56 ··· 56 size 1 1 7 7 2 4 4 14 28 28 2 2 2 2 2 14 14 2 2 2 4 4 4 8 ··· 8 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 Q16 D14 D14 D4×D7 D7×Q16 kernel D7×Q16 C8×D7 Dic28 C7⋊Q16 C7×Q16 Q8×D7 Dic7 D14 Q16 D7 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 1 1 3 4 3 6 3 6

Matrix representation of D7×Q16 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 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 0 14 0 0 8 62
,
 112 0 0 0 0 112 0 0 0 0 49 65 0 0 83 64
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,0,8,0,0,14,62],[112,0,0,0,0,112,0,0,0,0,49,83,0,0,65,64] >;

D7×Q16 in GAP, Magma, Sage, TeX

D_7\times Q_{16}
% in TeX

G:=Group("D7xQ16");
// GroupNames label

G:=SmallGroup(224,112);
// by ID

G=gap.SmallGroup(224,112);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,116,86,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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