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G = C84Q16order 128 = 27

1st semidirect product of C8 and Q16 acting via Q16/C8=C2

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

Aliases: C84Q16, C8.14D8, C82.5C2, C42.660C23, C4.4(C2×D8), C4.3(C2×Q16), (C2×C8).242D4, C82Q8.8C2, C2.7(C84D4), C4⋊Q16.6C2, C4⋊Q8.84C22, (C4×C8).395C22, C2.6(C4⋊Q16), C22.61(C41D4), (C2×C4).717(C2×D4), SmallGroup(128,445)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C84Q16
C1C2C22C2×C4C42C4×C8C82 — C84Q16
C1C22C42 — C84Q16
C1C22C42 — C84Q16
C1C22C22C42 — C84Q16

Generators and relations for C84Q16
 G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 208 in 96 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C8 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×8], C2×C8 [×6], Q16 [×8], C2×Q8 [×4], C4×C8, C4×C8 [×2], C2.D8 [×8], C4⋊Q8 [×4], C2×Q16 [×4], C82, C4⋊Q16 [×2], C82Q8 [×4], C84Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], Q16 [×8], C2×D4 [×3], C41D4, C2×D8 [×2], C2×Q16 [×4], C84D4, C4⋊Q16 [×2], C84Q16

Smallest permutation representation of C84Q16
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 17 35 28 96 42 85 101)(2 18 36 29 89 43 86 102)(3 19 37 30 90 44 87 103)(4 20 38 31 91 45 88 104)(5 21 39 32 92 46 81 97)(6 22 40 25 93 47 82 98)(7 23 33 26 94 48 83 99)(8 24 34 27 95 41 84 100)(9 62 123 50 71 111 75 119)(10 63 124 51 72 112 76 120)(11 64 125 52 65 105 77 113)(12 57 126 53 66 106 78 114)(13 58 127 54 67 107 79 115)(14 59 128 55 68 108 80 116)(15 60 121 56 69 109 73 117)(16 61 122 49 70 110 74 118)
(1 53 96 114)(2 52 89 113)(3 51 90 120)(4 50 91 119)(5 49 92 118)(6 56 93 117)(7 55 94 116)(8 54 95 115)(9 104 71 31)(10 103 72 30)(11 102 65 29)(12 101 66 28)(13 100 67 27)(14 99 68 26)(15 98 69 25)(16 97 70 32)(17 126 42 78)(18 125 43 77)(19 124 44 76)(20 123 45 75)(21 122 46 74)(22 121 47 73)(23 128 48 80)(24 127 41 79)(33 59 83 108)(34 58 84 107)(35 57 85 106)(36 64 86 105)(37 63 87 112)(38 62 88 111)(39 61 81 110)(40 60 82 109)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111222
type++++++-
imageC1C2C2C2D4D8Q16
kernelC84Q16C82C4⋊Q16C82Q8C2×C8C8C8
# reps11246816

Matrix representation of C84Q16 in GL4(𝔽17) generated by

3300
14300
0010
0001
,
14300
141400
001111
0030
,
11000
101600
00915
0078
G:=sub<GL(4,GF(17))| [3,14,0,0,3,3,0,0,0,0,1,0,0,0,0,1],[14,14,0,0,3,14,0,0,0,0,11,3,0,0,11,0],[1,10,0,0,10,16,0,0,0,0,9,7,0,0,15,8] >;

C84Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_4Q_{16}
% in TeX

G:=Group("C8:4Q16");
// GroupNames label

G:=SmallGroup(128,445);
// by ID

G=gap.SmallGroup(128,445);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,436,1123,136,2804,172]);
// Polycyclic

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

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