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

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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