C16:D7 - GroupNames

(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 C₁₁₂⋊₉C₄ D₁₄⋊C₁₆

68 conjugacy classes

class	1	2A	2B	4A	4B	4C	7A	7B	7C	8A	8B	8C	8D	8E	8F	14A	14B	14C	16A	16B	16C	16D	16E	16F	16G	16H	28A	···	28F	56A	···	56L	112A	···	112X
order	1	2	2	4	4	4	7	7	7	8	8	8	8	8	8	14	14	14	16	16	16	16	16	16	16	16	28	···	28	56	···	56	112	···	112
size	1	1	14	1	1	14	2	2	2	1	1	1	1	14	14	2	2	2	2	2	2	2	14	14	14	14	2	···	2	2	···	2	2	···	2

68 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+					+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	C₈	D₇	D₁₄	M₅(2)	C₄×D₇	C₈×D₇	C₁₆⋊D₇
kernel	C₁₆⋊D₇	C₇⋊C₁₆	C₁₁₂	C₈×D₇	C₇⋊C₈	C₄×D₇	Dic₇	D₁₄	C₁₆	C₈	C₇	C₄	C₂	C₁
# reps	1	1	1	1	2	2	4	4	3	3	4	6	12	24

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

9	10
11	32

36	14
40	19

22	32
40	19

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

G = C16⋊D7 order 224 = 25·7

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

G = C₁₆⋊D₇ order 224 = 2⁵·7

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