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G = C7⋊D16order 224 = 25·7

The semidirect product of C7 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C72D16, D81D7, D563C2, C28.3D4, C14.8D8, C8.4D14, C56.2C22, C7⋊C161C2, (C7×D8)⋊1C2, C2.4(D4⋊D7), C4.1(C7⋊D4), SmallGroup(224,32)

Series: Derived Chief Lower central Upper central

C1C56 — C7⋊D16
C1C7C14C28C56D56 — C7⋊D16
C7C14C28C56 — C7⋊D16
C1C2C4C8D8

Generators and relations for C7⋊D16
 G = < a,b,c | a7=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

8C2
56C2
4C22
28C22
8D7
8C14
2D4
14D4
4D14
4C2×C14
7C16
7D8
2D28
2C7×D4
7D16

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

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

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

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

C7⋊D16 is a maximal subgroup of   D7×D16  D8⋊D14  D112⋊C2  SD323D7  D8.D14  Q16⋊D14  C56.30C23
C7⋊D16 is a maximal quotient of   C8.4Dic14  C14.D16  C7⋊D32  D16.D7  C7⋊SD64  C7⋊Q64  C14.SD32

32 conjugacy classes

class 1 2A2B2C 4 7A7B7C8A8B14A14B14C14D···14I16A16B16C16D28A28B28C56A···56F
order122247778814141414···141616161628282856···56
size118562222222228···8141414144444···4

32 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2D4D7D8D14D16C7⋊D4D4⋊D7C7⋊D16
kernelC7⋊D16C7⋊C16D56C7×D8C28D8C14C8C7C4C2C1
# reps111113234636

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

0100
112900
0010
0001
,
112000
104100
00228
0010914
,
1000
911200
00112111
0001
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[112,104,0,0,0,1,0,0,0,0,22,109,0,0,8,14],[1,9,0,0,0,112,0,0,0,0,112,0,0,0,111,1] >;

C7⋊D16 in GAP, Magma, Sage, TeX

C_7\rtimes D_{16}
% in TeX

G:=Group("C7:D16");
// GroupNames label

G:=SmallGroup(224,32);
// by ID

G=gap.SmallGroup(224,32);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,218,116,122,579,297,69,6917]);
// Polycyclic

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

Export

Subgroup lattice of C7⋊D16 in TeX

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