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G = C163Q8order 128 = 27

2nd semidirect product of C16 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C163Q8, C8.9Q16, C4.5SD32, C42.342D4, (C2×C4).87D8, C4.7(C4⋊Q8), C8.20(C2×Q8), (C4×C16).14C2, (C2×C8).256D4, C4.11(C2×Q16), C164C4.7C2, C82Q8.15C2, C2.8(C82Q8), C2.17(C2×SD32), (C2×C8).552C23, (C4×C8).405C22, C22.138(C2×D8), (C2×C16).100C22, C2.D8.31C22, (C2×C4).820(C2×D4), SmallGroup(128,986)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C163Q8
C1C2C4C8C2×C8C2×C16C4×C16 — C163Q8
C1C2C4C2×C8 — C163Q8
C1C22C42C4×C8 — C163Q8
C1C2C2C2C2C4C4C2×C8 — C163Q8

Generators and relations for C163Q8
 G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a7, cbc-1=b-1 >

Subgroups: 152 in 64 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C4 [×4], C22, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C16 [×4], C42, C4⋊C4 [×6], C2×C8 [×2], C2×Q8 [×2], C4×C8, C2.D8 [×4], C2.D8 [×2], C2×C16 [×2], C4⋊Q8 [×2], C4×C16, C164C4 [×4], C82Q8 [×2], C163Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D8 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], SD32 [×4], C4⋊Q8, C2×D8, C2×Q16, C82Q8, C2×SD32 [×2], C163Q8

Smallest permutation representation of C163Q8
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)
(1 119 19 76)(2 120 20 77)(3 121 21 78)(4 122 22 79)(5 123 23 80)(6 124 24 65)(7 125 25 66)(8 126 26 67)(9 127 27 68)(10 128 28 69)(11 113 29 70)(12 114 30 71)(13 115 31 72)(14 116 32 73)(15 117 17 74)(16 118 18 75)(33 100 50 88)(34 101 51 89)(35 102 52 90)(36 103 53 91)(37 104 54 92)(38 105 55 93)(39 106 56 94)(40 107 57 95)(41 108 58 96)(42 109 59 81)(43 110 60 82)(44 111 61 83)(45 112 62 84)(46 97 63 85)(47 98 64 86)(48 99 49 87)
(1 84 19 112)(2 91 20 103)(3 82 21 110)(4 89 22 101)(5 96 23 108)(6 87 24 99)(7 94 25 106)(8 85 26 97)(9 92 27 104)(10 83 28 111)(11 90 29 102)(12 81 30 109)(13 88 31 100)(14 95 32 107)(15 86 17 98)(16 93 18 105)(33 115 50 72)(34 122 51 79)(35 113 52 70)(36 120 53 77)(37 127 54 68)(38 118 55 75)(39 125 56 66)(40 116 57 73)(41 123 58 80)(42 114 59 71)(43 121 60 78)(44 128 61 69)(45 119 62 76)(46 126 63 67)(47 117 64 74)(48 124 49 65)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J8A···8H16A···16P
order12224···444448···816···16
size11112···2161616162···22···2

38 irreducible representations

dim1111222222
type++++-++-+
imageC1C2C2C2Q8D4D4Q16D8SD32
kernelC163Q8C4×C16C164C4C82Q8C16C42C2×C8C8C2×C4C4
# reps11424114416

Matrix representation of C163Q8 in GL4(𝔽17) generated by

11000
0300
00716
0017
,
4000
01300
0010
0001
,
0100
16000
00114
0046
G:=sub<GL(4,GF(17))| [11,0,0,0,0,3,0,0,0,0,7,1,0,0,16,7],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,11,4,0,0,4,6] >;

C163Q8 in GAP, Magma, Sage, TeX

C_{16}\rtimes_3Q_8
% in TeX

G:=Group("C16:3Q8");
// GroupNames label

G:=SmallGroup(128,986);
// by ID

G=gap.SmallGroup(128,986);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,604,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations

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