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G = C8×Q16order 128 = 27

Direct product of C8 and Q16

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

Aliases: C8×Q16, C82.3C2, C42.634C23, C2.6(C8×D4), C82(Q8⋊C8), C8.10(C2×C8), C82(C81C8), C82(C2.D8), (C8×Q8).2C2, Q8.1(C2×C8), C2.1(C4×Q16), C4.9(C8○D4), C2.3(C8○D8), (C2×C8).219D4, Q8⋊C8.11C2, C4.9(C22×C8), C4.57(C2×Q16), C81C8.15C2, C2.D8.23C4, C82(Q8⋊C4), (C4×Q16).21C2, (C2×Q16).15C4, C22.79(C4×D4), C4.127(C4○D8), C4⋊C8.272C22, (C4×C8).368C22, Q8⋊C4.14C4, (C4×Q8).256C22, (C2×C8)(C4×Q16), C4⋊C4.128(C2×C4), (C2×C8).156(C2×C4), (C2×C4).1470(C2×D4), (C2×Q8).132(C2×C4), (C2×C4).495(C4○D4), (C2×C4).326(C22×C4), SmallGroup(128,309)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8×Q16
C1C2C22C2×C4C42C4×C8C8×Q8 — C8×Q16
C1C2C4 — C8×Q16
C1C2×C8C4×C8 — C8×Q16
C1C22C22C42 — C8×Q16

Generators and relations for C8×Q16
 G = < a,b,c | a8=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 120 in 82 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C4 [×4], C4 [×7], C22, C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Q8 [×2], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×3], Q16 [×4], C2×Q8 [×2], C4×C8 [×3], C4×C8 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C82, Q8⋊C8 [×2], C81C8, C4×Q16, C8×Q8 [×2], C8×Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], Q16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×Q16, C4○D8, C8×D4, C4×Q16, C8○D8, C8×Q16

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

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P8A···8H8I···8AB8AC···8AJ
order1222444444444···48···88···88···8
size1111111122224···41···12···24···4

56 irreducible representations

dim1111111111222222
type+++++++-
imageC1C2C2C2C2C2C4C4C4C8D4Q16C4○D4C8○D4C4○D8C8○D8
kernelC8×Q16C82Q8⋊C8C81C8C4×Q16C8×Q8Q8⋊C4C2.D8C2×Q16Q16C2×C8C8C2×C4C4C4C2
# reps11211242216242448

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

15000
01500
0040
0004
,
10900
2700
0003
00116
,
0200
9000
0091
0038
G:=sub<GL(4,GF(17))| [15,0,0,0,0,15,0,0,0,0,4,0,0,0,0,4],[10,2,0,0,9,7,0,0,0,0,0,11,0,0,3,6],[0,9,0,0,2,0,0,0,0,0,9,3,0,0,1,8] >;

C8×Q16 in GAP, Magma, Sage, TeX

C_8\times Q_{16}
% in TeX

G:=Group("C8xQ16");
// GroupNames label

G:=SmallGroup(128,309);
// by ID

G=gap.SmallGroup(128,309);
# by ID

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

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

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