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G = C85Q16order 128 = 27

2nd semidirect product of C8 and Q16 acting via Q16/C8=C2

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

Aliases: C85Q16, C82.6C2, C8.17SD16, C42.654C23, C4.1(C2×Q16), C4.2(C4○D8), (C2×C8).221D4, C83Q8.6C2, C82Q8.6C2, C4.3(C2×SD16), C2.7(C85D4), C4⋊Q16.5C2, C4⋊Q8.79C22, (C4×C8).428C22, C2.4(C4⋊Q16), C4.SD16.3C2, C2.7(C8.12D4), C22.55(C41D4), (C2×C4).711(C2×D4), SmallGroup(128,439)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C85Q16
C1C2C22C2×C4C42C4×C8C82 — C85Q16
C1C22C42 — C85Q16
C1C22C42 — C85Q16
C1C22C22C42 — C85Q16

Generators and relations for C85Q16
 G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a3, cbc-1=b-1 >

Subgroups: 192 in 90 conjugacy classes, 44 normal (16 characteristic)
C1, C2 [×3], C4 [×6], C4 [×4], C22, C8 [×8], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×8], C2×C8 [×6], Q16 [×4], C2×Q8 [×4], C4×C8 [×3], Q8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×2], C4⋊Q8 [×4], C2×Q16 [×2], C82, C4.SD16 [×2], C4⋊Q16, C83Q8 [×2], C82Q8, C85Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×4], Q16 [×4], C2×D4 [×3], C41D4, C2×SD16 [×2], C2×Q16 [×2], C4○D8 [×2], C85D4, C4⋊Q16, C8.12D4, C85Q16

Smallest permutation representation of C85Q16
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 63 23 73 47 38 16 54)(2 64 24 74 48 39 9 55)(3 57 17 75 41 40 10 56)(4 58 18 76 42 33 11 49)(5 59 19 77 43 34 12 50)(6 60 20 78 44 35 13 51)(7 61 21 79 45 36 14 52)(8 62 22 80 46 37 15 53)(25 91 123 88 120 66 98 109)(26 92 124 81 113 67 99 110)(27 93 125 82 114 68 100 111)(28 94 126 83 115 69 101 112)(29 95 127 84 116 70 102 105)(30 96 128 85 117 71 103 106)(31 89 121 86 118 72 104 107)(32 90 122 87 119 65 97 108)
(1 111 47 82)(2 106 48 85)(3 109 41 88)(4 112 42 83)(5 107 43 86)(6 110 44 81)(7 105 45 84)(8 108 46 87)(9 96 24 71)(10 91 17 66)(11 94 18 69)(12 89 19 72)(13 92 20 67)(14 95 21 70)(15 90 22 65)(16 93 23 68)(25 75 120 56)(26 78 113 51)(27 73 114 54)(28 76 115 49)(29 79 116 52)(30 74 117 55)(31 77 118 50)(32 80 119 53)(33 126 58 101)(34 121 59 104)(35 124 60 99)(36 127 61 102)(37 122 62 97)(38 125 63 100)(39 128 64 103)(40 123 57 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), (1,63,23,73,47,38,16,54)(2,64,24,74,48,39,9,55)(3,57,17,75,41,40,10,56)(4,58,18,76,42,33,11,49)(5,59,19,77,43,34,12,50)(6,60,20,78,44,35,13,51)(7,61,21,79,45,36,14,52)(8,62,22,80,46,37,15,53)(25,91,123,88,120,66,98,109)(26,92,124,81,113,67,99,110)(27,93,125,82,114,68,100,111)(28,94,126,83,115,69,101,112)(29,95,127,84,116,70,102,105)(30,96,128,85,117,71,103,106)(31,89,121,86,118,72,104,107)(32,90,122,87,119,65,97,108), (1,111,47,82)(2,106,48,85)(3,109,41,88)(4,112,42,83)(5,107,43,86)(6,110,44,81)(7,105,45,84)(8,108,46,87)(9,96,24,71)(10,91,17,66)(11,94,18,69)(12,89,19,72)(13,92,20,67)(14,95,21,70)(15,90,22,65)(16,93,23,68)(25,75,120,56)(26,78,113,51)(27,73,114,54)(28,76,115,49)(29,79,116,52)(30,74,117,55)(31,77,118,50)(32,80,119,53)(33,126,58,101)(34,121,59,104)(35,124,60,99)(36,127,61,102)(37,122,62,97)(38,125,63,100)(39,128,64,103)(40,123,57,98)>;

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

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

38 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J8A···8X
order12224···444448···8
size11112···2161616162···2

38 irreducible representations

dim1111112222
type+++++++-
imageC1C2C2C2C2C2D4SD16Q16C4○D8
kernelC85Q16C82C4.SD16C4⋊Q16C83Q8C82Q8C2×C8C8C8C4
# reps1121216888

Matrix representation of C85Q16 in GL4(𝔽17) generated by

8000
0200
0025
0008
,
15000
0800
0010
0001
,
0100
16000
00124
00115
G:=sub<GL(4,GF(17))| [8,0,0,0,0,2,0,0,0,0,2,0,0,0,5,8],[15,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,12,11,0,0,4,5] >;

C85Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_5Q_{16}
% in TeX

G:=Group("C8:5Q16");
// GroupNames label

G:=SmallGroup(128,439);
// by ID

G=gap.SmallGroup(128,439);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,100,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

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