Copied to
clipboard

G = C7×D16order 224 = 25·7

Direct product of C7 and D16

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

Aliases: C7×D16, C1123C2, C161C14, D81C14, C14.15D8, C28.36D4, C56.24C22, (C7×D8)⋊5C2, C4.1(C7×D4), C2.3(C7×D8), C8.2(C2×C14), SmallGroup(224,60)

Series: Derived Chief Lower central Upper central

C1C8 — C7×D16
C1C2C4C8C56C7×D8 — C7×D16
C1C2C4C8 — C7×D16
C1C14C28C56 — C7×D16

Generators and relations for C7×D16
 G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

8C2
8C2
4C22
4C22
8C14
8C14
2D4
2D4
4C2×C14
4C2×C14
2C7×D4
2C7×D4

Smallest permutation representation of C7×D16
On 112 points
Generators in S112
(1 43 105 57 32 77 85)(2 44 106 58 17 78 86)(3 45 107 59 18 79 87)(4 46 108 60 19 80 88)(5 47 109 61 20 65 89)(6 48 110 62 21 66 90)(7 33 111 63 22 67 91)(8 34 112 64 23 68 92)(9 35 97 49 24 69 93)(10 36 98 50 25 70 94)(11 37 99 51 26 71 95)(12 38 100 52 27 72 96)(13 39 101 53 28 73 81)(14 40 102 54 29 74 82)(15 41 103 55 30 75 83)(16 42 104 56 31 76 84)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)(33 36)(34 35)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)

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

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

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

C7×D16 is a maximal subgroup of   C7⋊D32  D16.D7  D8⋊D14  D163D7

77 conjugacy classes

class 1 2A2B2C 4 7A···7F8A8B14A···14F14G···14R16A16B16C16D28A···28F56A···56L112A···112X
order122247···78814···1414···141616161628···2856···56112···112
size118821···1221···18···822222···22···22···2

77 irreducible representations

dim111111222222
type++++++
imageC1C2C2C7C14C14D4D8D16C7×D4C7×D8C7×D16
kernelC7×D16C112C7×D8D16C16D8C28C14C7C4C2C1
# reps112661212461224

Matrix representation of C7×D16 in GL4(𝔽113) generated by

106000
010600
0010
0001
,
08200
625100
0018109
00418
,
513100
516200
0018109
0010995
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,1,0,0,0,0,1],[0,62,0,0,82,51,0,0,0,0,18,4,0,0,109,18],[51,51,0,0,31,62,0,0,0,0,18,109,0,0,109,95] >;

C7×D16 in GAP, Magma, Sage, TeX

C_7\times D_{16}
% in TeX

G:=Group("C7xD16");
// GroupNames label

G:=SmallGroup(224,60);
// by ID

G=gap.SmallGroup(224,60);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,361,2019,1017,165,5044,2530,88]);
// Polycyclic

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

Export

Subgroup lattice of C7×D16 in TeX

׿
×
𝔽