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G = C16⋊D7order 224 = 25·7

3rd semidirect product of C16 and D7 acting via D7/C7=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C163D7, D14.C8, C1125C2, Dic7.C8, C71M5(2), C8.20D14, C56.20C22, C7⋊C164C2, C7⋊C8.2C4, C2.3(C8×D7), C14.2(C2×C8), (C4×D7).2C4, (C8×D7).2C2, C4.17(C4×D7), C28.22(C2×C4), SmallGroup(224,4)

Series: Derived Chief Lower central Upper central

C1C14 — C16⋊D7
C1C7C14C28C56C8×D7 — C16⋊D7
C7C14 — C16⋊D7
C1C8C16

Generators and relations for C16⋊D7
 G = < a,b,c | a16=b7=c2=1, ab=ba, cac=a9, cbc=b-1 >

14C2
7C4
7C22
2D7
7C2×C4
7C8
7C2×C8
7C16
7M5(2)

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

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

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

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

C16⋊D7 is a maximal subgroup of   D28.4C8  D7×M5(2)  C16.12D14  D8⋊D14  D112⋊C2  SD32⋊D7  Q32⋊D7
C16⋊D7 is a maximal quotient of   Dic7⋊C16  C1129C4  D14⋊C16

68 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F14A14B14C16A16B16C16D16E16F16G16H28A···28F56A···56L112A···112X
order122444777888888141414161616161616161628···2856···56112···112
size11141114222111114142222222141414142···22···22···2

68 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8D7D14M5(2)C4×D7C8×D7C16⋊D7
kernelC16⋊D7C7⋊C16C112C8×D7C7⋊C8C4×D7Dic7D14C16C8C7C4C2C1
# reps1111224433461224

Matrix representation of C16⋊D7 in GL2(𝔽41) generated by

910
1132
,
3614
4019
,
2232
4019
G:=sub<GL(2,GF(41))| [9,11,10,32],[36,40,14,19],[22,40,32,19] >;

C16⋊D7 in GAP, Magma, Sage, TeX

C_{16}\rtimes D_7
% in TeX

G:=Group("C16:D7");
// GroupNames label

G:=SmallGroup(224,4);
// by ID

G=gap.SmallGroup(224,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,31,50,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^16=b^7=c^2=1,a*b=b*a,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C16⋊D7 in TeX

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