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G = C162Q8order 128 = 27

1st semidirect product of C16 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C162Q8, C4.5D16, C4.4Q32, C8.8Q16, C42.340D4, (C2×C4).86D8, C4.5(C4⋊Q8), C8.18(C2×Q8), C4.9(C2×Q16), (C4×C16).10C2, (C2×C8).255D4, C2.11(C2×D16), C2.11(C2×Q32), C163C4.6C2, C82Q8.14C2, C2.6(C82Q8), (C4×C8).404C22, (C2×C8).550C23, (C2×C16).86C22, C22.136(C2×D8), C2.D8.29C22, (C2×C4).818(C2×D4), SmallGroup(128,984)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C162Q8
C1C2C4C8C2×C8C2×C16C4×C16 — C162Q8
C1C2C4C2×C8 — C162Q8
C1C22C42C4×C8 — C162Q8
C1C2C2C2C2C4C4C2×C8 — C162Q8

Generators and relations for C162Q8
 G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 152 in 64 conjugacy classes, 40 normal (16 characteristic)
C1, C2 [×3], C4 [×2], C4 [×4], C4 [×4], C22, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C16 [×4], C42, C4⋊C4 [×6], C2×C8 [×2], C2×Q8 [×2], C4×C8, C2.D8 [×4], C2.D8 [×2], C2×C16 [×2], C4⋊Q8 [×2], C4×C16, C163C4 [×4], C82Q8 [×2], C162Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D8 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], D16 [×2], Q32 [×2], C4⋊Q8, C2×D8, C2×Q16, C82Q8, C2×D16, C2×Q32, C162Q8

Smallest permutation representation of C162Q8
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 66 90 24)(2 67 91 25)(3 68 92 26)(4 69 93 27)(5 70 94 28)(6 71 95 29)(7 72 96 30)(8 73 81 31)(9 74 82 32)(10 75 83 17)(11 76 84 18)(12 77 85 19)(13 78 86 20)(14 79 87 21)(15 80 88 22)(16 65 89 23)(33 56 107 117)(34 57 108 118)(35 58 109 119)(36 59 110 120)(37 60 111 121)(38 61 112 122)(39 62 97 123)(40 63 98 124)(41 64 99 125)(42 49 100 126)(43 50 101 127)(44 51 102 128)(45 52 103 113)(46 53 104 114)(47 54 105 115)(48 55 106 116)
(1 117 90 56)(2 116 91 55)(3 115 92 54)(4 114 93 53)(5 113 94 52)(6 128 95 51)(7 127 96 50)(8 126 81 49)(9 125 82 64)(10 124 83 63)(11 123 84 62)(12 122 85 61)(13 121 86 60)(14 120 87 59)(15 119 88 58)(16 118 89 57)(17 40 75 98)(18 39 76 97)(19 38 77 112)(20 37 78 111)(21 36 79 110)(22 35 80 109)(23 34 65 108)(24 33 66 107)(25 48 67 106)(26 47 68 105)(27 46 69 104)(28 45 70 103)(29 44 71 102)(30 43 72 101)(31 42 73 100)(32 41 74 99)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J8A···8H16A···16P
order12224···444448···816···16
size11112···2161616162···22···2

38 irreducible representations

dim11112222222
type++++-++-++-
imageC1C2C2C2Q8D4D4Q16D8D16Q32
kernelC162Q8C4×C16C163C4C82Q8C16C42C2×C8C8C2×C4C4C4
# reps11424114488

Matrix representation of C162Q8 in GL4(𝔽17) generated by

41100
6400
00136
001113
,
0100
16000
0010
0001
,
121200
12500
00114
0046
G:=sub<GL(4,GF(17))| [4,6,0,0,11,4,0,0,0,0,13,11,0,0,6,13],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,11,4,0,0,4,6] >;

C162Q8 in GAP, Magma, Sage, TeX

C_{16}\rtimes_2Q_8
% in TeX

G:=Group("C16:2Q8");
// GroupNames label

G:=SmallGroup(128,984);
// by ID

G=gap.SmallGroup(128,984);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,268,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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