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G = C8.7Q16order 128 = 27

3rd non-split extension by C8 of Q16 acting via Q16/C8=C2

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

Aliases: C8.7Q16, C82.4C2, C42.657C23, C4.2(C2×Q16), C4.5(C4○D8), (C2×C8).224D4, C82Q8.7C2, C4⋊Q8.82C22, (C4×C8).370C22, C2.5(C4⋊Q16), C4.SD16.4C2, C2.10(C8.12D4), C22.58(C41D4), (C2×C4).714(C2×D4), SmallGroup(128,442)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.7Q16
C1C2C22C2×C4C42C4×C8C82 — C8.7Q16
C1C22C42 — C8.7Q16
C1C22C42 — C8.7Q16
C1C22C22C42 — C8.7Q16

Generators and relations for C8.7Q16
 G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 176 in 84 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×8], C2×C8 [×6], C2×Q8 [×4], C4×C8, C4×C8 [×2], Q8⋊C4 [×8], C2.D8 [×4], C4⋊Q8 [×4], C82, C4.SD16 [×4], C82Q8 [×2], C8.7Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, Q16 [×4], C2×D4 [×3], C41D4, C2×Q16 [×2], C4○D8 [×4], C4⋊Q16, C8.12D4 [×2], C8.7Q16

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111222
type+++++-
imageC1C2C2C2D4Q16C4○D8
kernelC8.7Q16C82C4.SD16C82Q8C2×C8C8C4
# reps11426816

Matrix representation of C8.7Q16 in GL4(𝔽17) generated by

13000
0400
0090
0002
,
2000
0900
0010
00016
,
0100
16000
0001
0010
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,9,0,0,0,0,2],[2,0,0,0,0,9,0,0,0,0,1,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C8.7Q16 in GAP, Magma, Sage, TeX

C_8._7Q_{16}
% in TeX

G:=Group("C8.7Q16");
// GroupNames label

G:=SmallGroup(128,442);
// by ID

G=gap.SmallGroup(128,442);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,268,1123,136,2804,172]);
// 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=a^4*b^-1>;
// generators/relations

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