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## G = D7×C16order 224 = 25·7

### Direct product of C16 and D7

Aliases: D7×C16, C1124C2, D14.2C8, C8.19D14, Dic7.2C8, C56.19C22, C7⋊C166C2, C71(C2×C16), C7⋊C8.3C4, C2.1(C8×D7), C14.1(C2×C8), (C4×D7).4C4, (C8×D7).3C2, C4.16(C4×D7), C28.21(C2×C4), SmallGroup(224,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C16
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×C16
 Lower central C7 — D7×C16
 Upper central C1 — C16

Generators and relations for D7×C16
G = < a,b,c | a16=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C16
On 112 points
Generators in S112
(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 98 78 18 54 40 82)(2 99 79 19 55 41 83)(3 100 80 20 56 42 84)(4 101 65 21 57 43 85)(5 102 66 22 58 44 86)(6 103 67 23 59 45 87)(7 104 68 24 60 46 88)(8 105 69 25 61 47 89)(9 106 70 26 62 48 90)(10 107 71 27 63 33 91)(11 108 72 28 64 34 92)(12 109 73 29 49 35 93)(13 110 74 30 50 36 94)(14 111 75 31 51 37 95)(15 112 76 32 52 38 96)(16 97 77 17 53 39 81)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 97)(48 98)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)

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

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

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

D7×C16 is a maximal subgroup of   C32⋊D7  D28.4C8  C16.12D14  D163D7  SD323D7  Q323D7
D7×C16 is a maximal quotient of   C32⋊D7  Dic7⋊C16  D14⋊C16

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 D7 D14 C4×D7 C8×D7 D7×C16 kernel D7×C16 C7⋊C16 C112 C8×D7 C7⋊C8 C4×D7 Dic7 D14 D7 C16 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 3 3 6 12 24

Matrix representation of D7×C16 in GL2(𝔽113) generated by

 35 0 0 35
,
 0 1 112 9
,
 0 112 112 0
G:=sub<GL(2,GF(113))| [35,0,0,35],[0,112,1,9],[0,112,112,0] >;

D7×C16 in GAP, Magma, Sage, TeX

D_7\times C_{16}
% in TeX

G:=Group("D7xC16");
// GroupNames label

G:=SmallGroup(224,3);
// by ID

G=gap.SmallGroup(224,3);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,31,50,69,6917]);
// Polycyclic

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

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