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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 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C16 [×2], C16 [×3], C42, C42 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×Q8, C4×C8, C4×C8 [×2], C4⋊C8, C4⋊C8 [×2], C2×C16 [×2], C2×C16 [×2], C4×Q8, C4×C16, C165C4 [×2], C4⋊C16, C4⋊C16 [×2], C8×Q8, C164Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, M5(2) [×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 59 48 76)(2 60 33 77)(3 61 34 78)(4 62 35 79)(5 63 36 80)(6 64 37 65)(7 49 38 66)(8 50 39 67)(9 51 40 68)(10 52 41 69)(11 53 42 70)(12 54 43 71)(13 55 44 72)(14 56 45 73)(15 57 46 74)(16 58 47 75)(17 99 84 124)(18 100 85 125)(19 101 86 126)(20 102 87 127)(21 103 88 128)(22 104 89 113)(23 105 90 114)(24 106 91 115)(25 107 92 116)(26 108 93 117)(27 109 94 118)(28 110 95 119)(29 111 96 120)(30 112 81 121)(31 97 82 122)(32 98 83 123)
(1 28 48 95)(2 21 33 88)(3 30 34 81)(4 23 35 90)(5 32 36 83)(6 25 37 92)(7 18 38 85)(8 27 39 94)(9 20 40 87)(10 29 41 96)(11 22 42 89)(12 31 43 82)(13 24 44 91)(14 17 45 84)(15 26 46 93)(16 19 47 86)(49 125 66 100)(50 118 67 109)(51 127 68 102)(52 120 69 111)(53 113 70 104)(54 122 71 97)(55 115 72 106)(56 124 73 99)(57 117 74 108)(58 126 75 101)(59 119 76 110)(60 128 77 103)(61 121 78 112)(62 114 79 105)(63 123 80 98)(64 116 65 107)

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

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

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

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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