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G = C164Q8order 128 = 27

3rd semidirect product of C16 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C164Q8, C4.3M5(2), C4⋊C4.11C8, C4⋊C8.24C4, C2.6(C8×Q8), C4⋊C16.13C2, (C2×Q8).9C8, C4.48(C4×Q8), C8.44(C2×Q8), (C4×C16).18C2, (C4×Q8).19C4, (C8×Q8).16C2, C165C4.9C2, C2.9(D4○C16), C4.61(C8○D4), C8.105(C4○D4), (C2×C8).636C23, C42.173(C2×C4), (C2×C16).57C22, (C4×C8).378C22, C2.11(C2×M5(2)), C22.54(C22×C8), (C2×C4).31(C2×C8), (C2×C8).152(C2×C4), (C2×C4).621(C22×C4), SmallGroup(128,915)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C164Q8
C1C2C4C8C2×C8C2×C16C4×C16 — C164Q8
C1C22 — C164Q8
C1C2×C8 — C164Q8
C1C2C2C2C2C4C4C2×C8 — C164Q8

Generators and relations for C164Q8
 G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a9, cbc-1=b-1 >

Subgroups: 76 in 63 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C16, C16, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C16, C2×C16, C4×Q8, C4×C16, C165C4, C4⋊C16, C4⋊C16, C8×Q8, C164Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, C2×C8, C22×C4, C2×Q8, C4○D4, M5(2), C4×Q8, C22×C8, C8○D4, C8×Q8, C2×M5(2), D4○C16, C164Q8

Smallest permutation representation of C164Q8
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 34 80 52)(2 35 65 53)(3 36 66 54)(4 37 67 55)(5 38 68 56)(6 39 69 57)(7 40 70 58)(8 41 71 59)(9 42 72 60)(10 43 73 61)(11 44 74 62)(12 45 75 63)(13 46 76 64)(14 47 77 49)(15 48 78 50)(16 33 79 51)(17 86 109 123)(18 87 110 124)(19 88 111 125)(20 89 112 126)(21 90 97 127)(22 91 98 128)(23 92 99 113)(24 93 100 114)(25 94 101 115)(26 95 102 116)(27 96 103 117)(28 81 104 118)(29 82 105 119)(30 83 106 120)(31 84 107 121)(32 85 108 122)
(1 114 80 93)(2 123 65 86)(3 116 66 95)(4 125 67 88)(5 118 68 81)(6 127 69 90)(7 120 70 83)(8 113 71 92)(9 122 72 85)(10 115 73 94)(11 124 74 87)(12 117 75 96)(13 126 76 89)(14 119 77 82)(15 128 78 91)(16 121 79 84)(17 35 109 53)(18 44 110 62)(19 37 111 55)(20 46 112 64)(21 39 97 57)(22 48 98 50)(23 41 99 59)(24 34 100 52)(25 43 101 61)(26 36 102 54)(27 45 103 63)(28 38 104 56)(29 47 105 49)(30 40 106 58)(31 33 107 51)(32 42 108 60)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12224444444444448···88888888816···1616···16
size11111111222244441···1222244442···24···4

56 irreducible representations

dim11111111122222
type+++++-
imageC1C2C2C2C2C4C4C8C8Q8C4○D4M5(2)C8○D4D4○C16
kernelC164Q8C4×C16C165C4C4⋊C16C8×Q8C4⋊C8C4×Q8C4⋊C4C2×Q8C16C8C4C4C2
# reps112316212422848

Matrix representation of C164Q8 in GL4(𝔽17) generated by

0900
13000
0009
0040
,
16000
01600
00011
0030
,
111100
3600
00169
00131
G:=sub<GL(4,GF(17))| [0,13,0,0,9,0,0,0,0,0,0,4,0,0,9,0],[16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,0],[11,3,0,0,11,6,0,0,0,0,16,13,0,0,9,1] >;

C164Q8 in GAP, Magma, Sage, TeX

C_{16}\rtimes_4Q_8
% in TeX

G:=Group("C16:4Q8");
// GroupNames label

G:=SmallGroup(128,915);
// by ID

G=gap.SmallGroup(128,915);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1430,142,102,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations

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