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G = C7⋊C16order 112 = 24·7

The semidirect product of C7 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7⋊C16, C14.C8, C8.2D7, C28.2C4, C56.2C2, C4.2Dic7, C2.(C7⋊C8), SmallGroup(112,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C16
C1C7C14C28C56 — C7⋊C16
C7 — C7⋊C16
C1C8

Generators and relations for C7⋊C16
 G = < a,b | a7=b16=1, bab-1=a-1 >

7C16

Smallest permutation representation of C7⋊C16
Regular action on 112 points
Generators in S112
(1 83 46 68 108 19 55)(2 56 20 109 69 47 84)(3 85 48 70 110 21 57)(4 58 22 111 71 33 86)(5 87 34 72 112 23 59)(6 60 24 97 73 35 88)(7 89 36 74 98 25 61)(8 62 26 99 75 37 90)(9 91 38 76 100 27 63)(10 64 28 101 77 39 92)(11 93 40 78 102 29 49)(12 50 30 103 79 41 94)(13 95 42 80 104 31 51)(14 52 32 105 65 43 96)(15 81 44 66 106 17 53)(16 54 18 107 67 45 82)
(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)

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

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

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

C7⋊C16 is a maximal subgroup of
D7×C16  C16⋊D7  C28.C8  C7⋊D16  D8.D7  C7⋊SD32  C7⋊Q32  C7⋊C48  C21⋊C16
C7⋊C16 is a maximal quotient of
C7⋊C32  C21⋊C16

40 conjugacy classes

class 1  2 4A4B7A7B7C8A8B8C8D14A14B14C16A···16H28A···28F56A···56L
order1244777888814141416···1628···2856···56
size111122211112227···72···22···2

40 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D7Dic7C7⋊C8C7⋊C16
kernelC7⋊C16C56C28C14C7C8C4C2C1
# reps1124833612

Matrix representation of C7⋊C16 in GL2(𝔽41) generated by

4040
3231
,
356
156
G:=sub<GL(2,GF(41))| [40,32,40,31],[35,15,6,6] >;

C7⋊C16 in GAP, Magma, Sage, TeX

C_7\rtimes C_{16}
% in TeX

G:=Group("C7:C16");
// GroupNames label

G:=SmallGroup(112,1);
// by ID

G=gap.SmallGroup(112,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,10,26,42,2404]);
// Polycyclic

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

Export

Subgroup lattice of C7⋊C16 in TeX

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