Copied to
clipboard

G = D7×C16order 224 = 25·7

Direct product of C16 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C7 — D7×C16
C1C7C14C28C56C8×D7 — D7×C16
C7 — D7×C16
C1C16

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

7C2
7C2
7C22
7C4
7C2×C4
7C8
7C2×C8
7C16
7C2×C16

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 2A2B2C4A4B4C4D7A7B7C8A8B8C8D8E8F8G8H14A14B14C16A···16H16I···16P28A···28F56A···56L112A···112X
order122244447778888888814141416···1616···1628···2856···56112···112
size11771177222111177772221···17···72···22···22···2

80 irreducible representations

dim11111111122222
type++++++
imageC1C2C2C2C4C4C8C8C16D7D14C4×D7C8×D7D7×C16
kernelD7×C16C7⋊C16C112C8×D7C7⋊C8C4×D7Dic7D14D7C16C8C4C2C1
# reps11112244163361224

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

350
035
,
01
1129
,
0112
1120
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

Export

Subgroup lattice of D7×C16 in TeX

׿
×
𝔽