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G = C8:6Q16order 128 = 27

3rd semidirect product of C8 and Q16 acting via Q16/C8=C2

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

Aliases: C8:6Q16, C82.7C2, C8.8M4(2), C42.648C23, Q8:C8.9C2, C2.5(C4xQ16), (C2xC8).286D4, (C2xQ16).8C4, (C4xQ16).3C2, C4.59(C2xQ16), C2.D8.14C4, C8:1C8.14C2, C4.15(C8oD4), C2.12(C8oD8), C2.8(C8:6D4), Q8:C4.2C4, C8:4Q8.13C2, C4.9(C2xM4(2)), C4.134(C4oD8), C4:C8.226C22, (C4xC8).393C22, C22.139(C4xD4), (C4xQ8).14C22, C4:C4.61(C2xC4), (C2xC8).170(C2xC4), (C2xQ8).54(C2xC4), (C2xC4).1484(C2xD4), (C2xC4).509(C4oD4), (C2xC4).340(C22xC4), SmallGroup(128,323)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C8:6Q16
C1C2C22C2xC4C42C4xC8C8:4Q8 — C8:6Q16
C1C2C2xC4 — C8:6Q16
C1C2xC4C4xC8 — C8:6Q16
C1C22C22C42 — C8:6Q16

Generators and relations for C8:6Q16
 G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a5, cbc-1=b-1 >

Subgroups: 120 in 76 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2xC4, C2xC4, Q8, C42, C42, C4:C4, C4:C4, C2xC8, C2xC8, Q16, C2xQ8, C4xC8, C8:C4, Q8:C4, C4:C8, C4:C8, C2.D8, C4xQ8, C2xQ16, C82, Q8:C8, C8:1C8, C4xQ16, C8:4Q8, C8:6Q16
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), Q16, C22xC4, C2xD4, C4oD4, C4xD4, C2xM4(2), C8oD4, C2xQ16, C4oD8, C8:6D4, C4xQ16, C8oD8, C8:6Q16

Smallest permutation representation of C8:6Q16
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 22 34 117 25 111 10 122)(2 23 35 118 26 112 11 123)(3 24 36 119 27 105 12 124)(4 17 37 120 28 106 13 125)(5 18 38 113 29 107 14 126)(6 19 39 114 30 108 15 127)(7 20 40 115 31 109 16 128)(8 21 33 116 32 110 9 121)(41 94 56 76 70 83 59 98)(42 95 49 77 71 84 60 99)(43 96 50 78 72 85 61 100)(44 89 51 79 65 86 62 101)(45 90 52 80 66 87 63 102)(46 91 53 73 67 88 64 103)(47 92 54 74 68 81 57 104)(48 93 55 75 69 82 58 97)
(1 56 25 59)(2 53 26 64)(3 50 27 61)(4 55 28 58)(5 52 29 63)(6 49 30 60)(7 54 31 57)(8 51 32 62)(9 65 33 44)(10 70 34 41)(11 67 35 46)(12 72 36 43)(13 69 37 48)(14 66 38 45)(15 71 39 42)(16 68 40 47)(17 93 106 82)(18 90 107 87)(19 95 108 84)(20 92 109 81)(21 89 110 86)(22 94 111 83)(23 91 112 88)(24 96 105 85)(73 118 103 123)(74 115 104 128)(75 120 97 125)(76 117 98 122)(77 114 99 127)(78 119 100 124)(79 116 101 121)(80 113 102 126)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L8A···8X8Y8Z8AA8AB
order12224444444444448···88888
size11111111222288882···28888

44 irreducible representations

dim1111111112222222
type+++++++-
imageC1C2C2C2C2C2C4C4C4D4M4(2)Q16C4oD4C8oD4C4oD8C8oD8
kernelC8:6Q16C82Q8:C8C8:1C8C4xQ16C8:4Q8Q8:C4C2.D8C2xQ16C2xC8C8C8C2xC4C4C4C2
# reps1121124222442448

Matrix representation of C8:6Q16 in GL4(F17) generated by

0900
8000
0001
0040
,
141400
31400
00015
0090
,
1700
71600
00118
0026
G:=sub<GL(4,GF(17))| [0,8,0,0,9,0,0,0,0,0,0,4,0,0,1,0],[14,3,0,0,14,14,0,0,0,0,0,9,0,0,15,0],[1,7,0,0,7,16,0,0,0,0,11,2,0,0,8,6] >;

C8:6Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_6Q_{16}
% in TeX

G:=Group("C8:6Q16");
// GroupNames label

G:=SmallGroup(128,323);
// by ID

G=gap.SmallGroup(128,323);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,268,1123,570,136,172]);
// Polycyclic

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

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