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G = C86Q16order 128 = 27

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

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

Aliases: C86Q16, C82.7C2, C8.8M4(2), C42.648C23, Q8⋊C8.9C2, C2.5(C4×Q16), (C2×C8).286D4, (C2×Q16).8C4, (C4×Q16).3C2, C4.59(C2×Q16), C2.D8.14C4, C81C8.14C2, C4.15(C8○D4), C2.12(C8○D8), C2.8(C86D4), Q8⋊C4.2C4, C84Q8.13C2, C4.9(C2×M4(2)), C4.134(C4○D8), C4⋊C8.226C22, (C4×C8).393C22, C22.139(C4×D4), (C4×Q8).14C22, C4⋊C4.61(C2×C4), (C2×C8).170(C2×C4), (C2×Q8).54(C2×C4), (C2×C4).1484(C2×D4), (C2×C4).509(C4○D4), (C2×C4).340(C22×C4), SmallGroup(128,323)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C86Q16
C1C2C22C2×C4C42C4×C8C84Q8 — C86Q16
C1C2C2×C4 — C86Q16
C1C2×C4C4×C8 — C86Q16
C1C22C22C42 — C86Q16

Generators and relations for C86Q16
 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 [×3], C4 [×4], C4 [×5], C22, C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×3], Q16 [×2], C2×Q8 [×2], C4×C8 [×3], C8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C82, Q8⋊C8 [×2], C81C8, C4×Q16, C84Q8 [×2], C86Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], Q16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×Q16, C4○D8, C86D4, C4×Q16, C8○D8, C86Q16

Smallest permutation representation of C86Q16
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 13 107 29 122)(2 23 35 118 14 108 30 123)(3 24 36 119 15 109 31 124)(4 17 37 120 16 110 32 125)(5 18 38 113 9 111 25 126)(6 19 39 114 10 112 26 127)(7 20 40 115 11 105 27 128)(8 21 33 116 12 106 28 121)(41 90 54 76 66 81 63 102)(42 91 55 77 67 82 64 103)(43 92 56 78 68 83 57 104)(44 93 49 79 69 84 58 97)(45 94 50 80 70 85 59 98)(46 95 51 73 71 86 60 99)(47 96 52 74 72 87 61 100)(48 89 53 75 65 88 62 101)
(1 50 13 59)(2 55 14 64)(3 52 15 61)(4 49 16 58)(5 54 9 63)(6 51 10 60)(7 56 11 57)(8 53 12 62)(17 93 110 84)(18 90 111 81)(19 95 112 86)(20 92 105 83)(21 89 106 88)(22 94 107 85)(23 91 108 82)(24 96 109 87)(25 66 38 41)(26 71 39 46)(27 68 40 43)(28 65 33 48)(29 70 34 45)(30 67 35 42)(31 72 36 47)(32 69 37 44)(73 114 99 127)(74 119 100 124)(75 116 101 121)(76 113 102 126)(77 118 103 123)(78 115 104 128)(79 120 97 125)(80 117 98 122)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim1111111112222222
type+++++++-
imageC1C2C2C2C2C2C4C4C4D4M4(2)Q16C4○D4C8○D4C4○D8C8○D8
kernelC86Q16C82Q8⋊C8C81C8C4×Q16C84Q8Q8⋊C4C2.D8C2×Q16C2×C8C8C8C2×C4C4C4C2
# reps1121124222442448

Matrix representation of C86Q16 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] >;

C86Q16 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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