C8:Q16 - GroupNames

G = C₈⋊Q₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and Q₁₆ acting via Q₁₆/C₄=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₈⋊₁Q₁₆, C₄².₂₄₃C₂³, C₄⋊C₄.₆₅D₄, (C₂×C₈).₉₅D₄, (C₂×Q₈).₅₈D₄, C₄.₄₂(C₂×Q₁₆), C₈⋊₃Q₈.₁C₂, C₈⋊₄Q₈.₂C₂, C₈⋊₁C₈.₁₁C₂, C₄⋊C₈.₃₁C₂², C₄.Q₁₆.₆C₂, C₄⋊₂Q₁₆.₇C₂, C₄⋊Q₈.₆₄C₂², C₂.₁₁(C₈⋊D₄), C₄.₄₃(C₈⋊C₂²), (C₄×C₈).₁₄₆C₂², C₂.₉(C₄⋊₂Q₁₆), C₄.₁₀D₈.₇C₂, C₄.₆Q₁₆.₅C₂, (C₄×Q₈).₄₆C₂², C₄.₇₃(C₈.C₂²), C₂.₁₉(D₄.₃D₄), C₂².₂₀₄(C₄⋊D₄), (C₂×C₄).₂₈(C₄○D₄), (C₂×C₄).₁₂₇₈(C₂×D₄), SmallGroup(128,424)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₄² — C₈⋊Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₈⋊Q₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊Q₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊Q₁₆

Generators and relations for C₈⋊Q₁₆
G = < a,b,c | a⁸=b⁸=1, c²=b⁴, bab^-1=a^-1, cac^-1=a⁵, cbc^-1=b^-1 >

Subgroups: 152 in 73 conjugacy classes, 34 normal (32 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₄.₁₀D₈, C₄.₆Q₁₆, C₈⋊₁C₈, C₈⋊₄Q₈, C₄⋊₂Q₁₆, C₄.Q₁₆, C₈⋊₃Q₈, C₈⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×Q₁₆, C₈⋊C₂², C₈.C₂², C₄⋊₂Q₁₆, C₈⋊D₄, D₄.₃D₄, C₈⋊Q₁₆

Character table of C₈⋊Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	2	2	4	8	8	16	16	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₉	2	2	2	2	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2	-2	0	√2	-√2	-√2	0	0	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	-√2	-√2	√2	0	0	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2	-2	0	-√2	√2	√2	0	0	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2	2	0	√2	√2	-√2	0	0	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	complex lifted from C₄○D₄
ρ₁₈	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	complex lifted from C₄○D₄
ρ₁₉	4	-4	4	-4	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₀	4	-4	-4	4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₁	4	-4	4	-4	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	2√-2	0	0	-2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	0	-2√-2	0	0	2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄

Smallest permutation representation of C₈⋊Q₁₆
►Regular action on 128 points

Generators in S₁₂₈

(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)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)

(1 41 78 118 30 81 57 14)(2 48 79 117 31 88 58 13)(3 47 80 116 32 87 59 12)(4 46 73 115 25 86 60 11)(5 45 74 114 26 85 61 10)(6 44 75 113 27 84 62 9)(7 43 76 120 28 83 63 16)(8 42 77 119 29 82 64 15)(17 90 69 105 56 34 102 124)(18 89 70 112 49 33 103 123)(19 96 71 111 50 40 104 122)(20 95 72 110 51 39 97 121)(21 94 65 109 52 38 98 128)(22 93 66 108 53 37 99 127)(23 92 67 107 54 36 100 126)(24 91 68 106 55 35 101 125)

(1 18 30 49)(2 23 31 54)(3 20 32 51)(4 17 25 56)(5 22 26 53)(6 19 27 50)(7 24 28 55)(8 21 29 52)(9 96 113 40)(10 93 114 37)(11 90 115 34)(12 95 116 39)(13 92 117 36)(14 89 118 33)(15 94 119 38)(16 91 120 35)(41 123 81 112)(42 128 82 109)(43 125 83 106)(44 122 84 111)(45 127 85 108)(46 124 86 105)(47 121 87 110)(48 126 88 107)(57 70 78 103)(58 67 79 100)(59 72 80 97)(60 69 73 102)(61 66 74 99)(62 71 75 104)(63 68 76 101)(64 65 77 98)

G:=sub<Sym(128)| (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)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,41,78,118,30,81,57,14)(2,48,79,117,31,88,58,13)(3,47,80,116,32,87,59,12)(4,46,73,115,25,86,60,11)(5,45,74,114,26,85,61,10)(6,44,75,113,27,84,62,9)(7,43,76,120,28,83,63,16)(8,42,77,119,29,82,64,15)(17,90,69,105,56,34,102,124)(18,89,70,112,49,33,103,123)(19,96,71,111,50,40,104,122)(20,95,72,110,51,39,97,121)(21,94,65,109,52,38,98,128)(22,93,66,108,53,37,99,127)(23,92,67,107,54,36,100,126)(24,91,68,106,55,35,101,125), (1,18,30,49)(2,23,31,54)(3,20,32,51)(4,17,25,56)(5,22,26,53)(6,19,27,50)(7,24,28,55)(8,21,29,52)(9,96,113,40)(10,93,114,37)(11,90,115,34)(12,95,116,39)(13,92,117,36)(14,89,118,33)(15,94,119,38)(16,91,120,35)(41,123,81,112)(42,128,82,109)(43,125,83,106)(44,122,84,111)(45,127,85,108)(46,124,86,105)(47,121,87,110)(48,126,88,107)(57,70,78,103)(58,67,79,100)(59,72,80,97)(60,69,73,102)(61,66,74,99)(62,71,75,104)(63,68,76,101)(64,65,77,98)>;

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)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,41,78,118,30,81,57,14)(2,48,79,117,31,88,58,13)(3,47,80,116,32,87,59,12)(4,46,73,115,25,86,60,11)(5,45,74,114,26,85,61,10)(6,44,75,113,27,84,62,9)(7,43,76,120,28,83,63,16)(8,42,77,119,29,82,64,15)(17,90,69,105,56,34,102,124)(18,89,70,112,49,33,103,123)(19,96,71,111,50,40,104,122)(20,95,72,110,51,39,97,121)(21,94,65,109,52,38,98,128)(22,93,66,108,53,37,99,127)(23,92,67,107,54,36,100,126)(24,91,68,106,55,35,101,125), (1,18,30,49)(2,23,31,54)(3,20,32,51)(4,17,25,56)(5,22,26,53)(6,19,27,50)(7,24,28,55)(8,21,29,52)(9,96,113,40)(10,93,114,37)(11,90,115,34)(12,95,116,39)(13,92,117,36)(14,89,118,33)(15,94,119,38)(16,91,120,35)(41,123,81,112)(42,128,82,109)(43,125,83,106)(44,122,84,111)(45,127,85,108)(46,124,86,105)(47,121,87,110)(48,126,88,107)(57,70,78,103)(58,67,79,100)(59,72,80,97)(60,69,73,102)(61,66,74,99)(62,71,75,104)(63,68,76,101)(64,65,77,98) );

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),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,41,78,118,30,81,57,14),(2,48,79,117,31,88,58,13),(3,47,80,116,32,87,59,12),(4,46,73,115,25,86,60,11),(5,45,74,114,26,85,61,10),(6,44,75,113,27,84,62,9),(7,43,76,120,28,83,63,16),(8,42,77,119,29,82,64,15),(17,90,69,105,56,34,102,124),(18,89,70,112,49,33,103,123),(19,96,71,111,50,40,104,122),(20,95,72,110,51,39,97,121),(21,94,65,109,52,38,98,128),(22,93,66,108,53,37,99,127),(23,92,67,107,54,36,100,126),(24,91,68,106,55,35,101,125)], [(1,18,30,49),(2,23,31,54),(3,20,32,51),(4,17,25,56),(5,22,26,53),(6,19,27,50),(7,24,28,55),(8,21,29,52),(9,96,113,40),(10,93,114,37),(11,90,115,34),(12,95,116,39),(13,92,117,36),(14,89,118,33),(15,94,119,38),(16,91,120,35),(41,123,81,112),(42,128,82,109),(43,125,83,106),(44,122,84,111),(45,127,85,108),(46,124,86,105),(47,121,87,110),(48,126,88,107),(57,70,78,103),(58,67,79,100),(59,72,80,97),(60,69,73,102),(61,66,74,99),(62,71,75,104),(63,68,76,101),(64,65,77,98)]])

Matrix representation of C₈⋊Q₁₆ ►in GL₈(𝔽₁₇)

0	14	8	9	0	0	0	0
3	0	9	9	0	0	0	0
9	8	0	14	0	0	0	0
8	8	3	0	0	0	0	0
0	0	0	0	0	0	0	12
0	0	0	0	0	0	5	0
0	0	0	0	0	10	0	7
0	0	0	0	7	0	10	0

11	11	2	5	0	0	0	0
6	11	5	15	0	0	0	0
12	2	6	11	0	0	0	0
2	5	6	6	0	0	0	0
0	0	0	0	2	4	12	6
0	0	0	0	13	2	11	12
0	0	0	0	11	3	15	13
0	0	0	0	14	11	4	15

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	0	0	14	5	3	0
0	0	0	0	5	3	0	14
0	0	0	0	11	0	3	5
0	0	0	0	0	6	5	14

G:=sub<GL(8,GF(17))| [0,3,9,8,0,0,0,0,14,0,8,8,0,0,0,0,8,9,0,3,0,0,0,0,9,9,14,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,10,0,0,0,0,0,0,5,0,10,0,0,0,0,12,0,7,0],[11,6,12,2,0,0,0,0,11,11,2,5,0,0,0,0,2,5,6,6,0,0,0,0,5,15,11,6,0,0,0,0,0,0,0,0,2,13,11,14,0,0,0,0,4,2,3,11,0,0,0,0,12,11,15,4,0,0,0,0,6,12,13,15],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,5,11,0,0,0,0,0,5,3,0,6,0,0,0,0,3,0,3,5,0,0,0,0,0,14,5,14] >;

C₈⋊Q₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes Q_{16}

% in TeX

G:=Group("C8:Q16");

// GroupNames label

G:=SmallGroup(128,424);

// by ID

G=gap.SmallGroup(128,424);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,387,352,1123,136,2804,718,172]);

// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=b^-1>;

// generators/relations

Export

Character table of C₈⋊Q₁₆ in TeX

0	14	8	9	0	0	0	0
3	0	9	9	0	0	0	0
9	8	0	14	0	0	0	0
8	8	3	0	0	0	0	0
0	0	0	0	0	0	0	12
0	0	0	0	0	0	5	0
0	0	0	0	0	10	0	7
0	0	0	0	7	0	10	0

11	11	2	5	0	0	0	0
6	11	5	15	0	0	0	0
12	2	6	11	0	0	0	0
2	5	6	6	0	0	0	0
0	0	0	0	2	4	12	6
0	0	0	0	13	2	11	12
0	0	0	0	11	3	15	13
0	0	0	0	14	11	4	15

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	0	0	14	5	3	0
0	0	0	0	5	3	0	14
0	0	0	0	11	0	3	5
0	0	0	0	0	6	5	14

0	14	8	9	0	0	0	0
3	0	9	9	0	0	0	0
9	8	0	14	0	0	0	0
8	8	3	0	0	0	0	0
0	0	0	0	0	0	0	12
0	0	0	0	0	0	5	0
0	0	0	0	0	10	0	7
0	0	0	0	7	0	10	0

11	11	2	5	0	0	0	0
6	11	5	15	0	0	0	0
12	2	6	11	0	0	0	0
2	5	6	6	0	0	0	0
0	0	0	0	2	4	12	6
0	0	0	0	13	2	11	12
0	0	0	0	11	3	15	13
0	0	0	0	14	11	4	15

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	0	0	14	5	3	0
0	0	0	0	5	3	0	14
0	0	0	0	11	0	3	5
0	0	0	0	0	6	5	14

G = C8⋊Q16 order 128 = 27

1st semidirect product of C8 and Q16 acting via Q16/C4=C22

G = C₈⋊Q₁₆ order 128 = 2⁷

1^st semidirect product of C₈ and Q₁₆ acting via Q₁₆/C₄=C₂²

0	14	8	9	0	0	0	0
3	0	9	9	0	0	0	0
9	8	0	14	0	0	0	0
8	8	3	0	0	0	0	0
0	0	0	0	0	0	0	12
0	0	0	0	0	0	5	0
0	0	0	0	0	10	0	7
0	0	0	0	7	0	10	0

11	11	2	5	0	0	0	0
6	11	5	15	0	0	0	0
12	2	6	11	0	0	0	0
2	5	6	6	0	0	0	0
0	0	0	0	2	4	12	6
0	0	0	0	13	2	11	12
0	0	0	0	11	3	15	13
0	0	0	0	14	11	4	15

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	0	0	14	5	3	0
0	0	0	0	5	3	0	14
0	0	0	0	11	0	3	5
0	0	0	0	0	6	5	14