C7:D16 - GroupNames

(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 SD₃₂⋊₃D₇ 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	2A	2B	2C	4	7A	7B	7C	8A	8B	14A	14B	14C	14D	···	14I	16A	16B	16C	16D	28A	28B	28C	56A	···	56F
order	1	2	2	2	4	7	7	7	8	8	14	14	14	14	···	14	16	16	16	16	28	28	28	56	···	56
size	1	1	8	56	2	2	2	2	2	2	2	2	2	8	···	8	14	14	14	14	4	4	4	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₈

D₁₄

D₁₆

C₇⋊D₄

D₄⋊D₇

C₇⋊D₁₆

kernel

C₇⋊D₁₆

C₇⋊C₁₆

D₅₆

C₇×D₈

C₂₈

D₈

C₁₄

C₈

C₇

C₄

C₂

C₁

# reps

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

0	1	0	0
112	9	0	0
0	0	1	0
0	0	0	1

112	0	0	0
104	1	0	0
0	0	22	8
0	0	109	14

1	0	0	0
9	112	0	0
0	0	112	111
0	0	0	1

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

G = C7⋊D16 order 224 = 25·7

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

G = C₇⋊D₁₆ order 224 = 2⁵·7

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