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G = C89Q16order 128 = 27

3rd semidirect product of C8 and Q16 acting via Q16/Q8=C2

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

Aliases: C89Q16, Q8.1M4(2), C42.641C23, Q8⋊C8.2C2, C2.4(C4×Q16), C8⋊C8.7C2, (C2×C8).380D4, (C2×Q16).6C4, (C8×Q8).14C2, (C4×Q16).2C2, C4.58(C2×Q16), C2.D8.10C4, C81C8.13C2, C4.33(C8○D4), (C4×C8).39C22, Q8⋊C4.3C4, C84Q8.11C2, C4.131(C4○D8), C2.9(C8.26D4), C2.11(C89D4), C4⋊C8.223C22, C22.132(C4×D4), C4.27(C2×M4(2)), (C4×Q8).10C22, (C2×C8).31(C2×C4), C4⋊C4.135(C2×C4), (C2×C4).1477(C2×D4), (C2×Q8).136(C2×C4), (C2×C4).502(C4○D4), (C2×C4).333(C22×C4), SmallGroup(128,316)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C89Q16
C1C2C22C2×C4C42C4×C8C8×Q8 — C89Q16
C1C2C2×C4 — C89Q16
C1C2×C4C4×C8 — C89Q16
C1C22C22C42 — C89Q16

Generators and relations for C89Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a5, ac=ca, cbc-1=b-1 >

Subgroups: 120 in 76 conjugacy classes, 44 normal (40 characteristic)
C1, C2 [×3], C4 [×4], C4 [×6], C22, C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×3], Q16 [×2], C2×Q8 [×2], C4×C8 [×3], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C8⋊C8, Q8⋊C8 [×2], C81C8, C4×Q16, C8×Q8, C84Q8, C89Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], Q16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×Q16, C4○D8, C89D4, C4×Q16, C8.26D4, C89Q16

Smallest permutation representation of C89Q16
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)
(1 110 12 126 31 20 37 113)(2 107 13 123 32 17 38 118)(3 112 14 128 25 22 39 115)(4 109 15 125 26 19 40 120)(5 106 16 122 27 24 33 117)(6 111 9 127 28 21 34 114)(7 108 10 124 29 18 35 119)(8 105 11 121 30 23 36 116)(41 95 51 79 65 82 60 100)(42 92 52 76 66 87 61 97)(43 89 53 73 67 84 62 102)(44 94 54 78 68 81 63 99)(45 91 55 75 69 86 64 104)(46 96 56 80 70 83 57 101)(47 93 49 77 71 88 58 98)(48 90 50 74 72 85 59 103)
(1 63 31 54)(2 64 32 55)(3 57 25 56)(4 58 26 49)(5 59 27 50)(6 60 28 51)(7 61 29 52)(8 62 30 53)(9 65 34 41)(10 66 35 42)(11 67 36 43)(12 68 37 44)(13 69 38 45)(14 70 39 46)(15 71 40 47)(16 72 33 48)(17 91 107 86)(18 92 108 87)(19 93 109 88)(20 94 110 81)(21 95 111 82)(22 96 112 83)(23 89 105 84)(24 90 106 85)(73 116 102 121)(74 117 103 122)(75 118 104 123)(76 119 97 124)(77 120 98 125)(78 113 99 126)(79 114 100 127)(80 115 101 128)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E···8R8S8T
order12224444444444444488888···888
size11111111222244448822224···488

38 irreducible representations

dim11111111112222224
type++++++++-
imageC1C2C2C2C2C2C2C4C4C4D4Q16C4○D4M4(2)C8○D4C4○D8C8.26D4
kernelC89Q16C8⋊C8Q8⋊C8C81C8C4×Q16C8×Q8C84Q8Q8⋊C4C2.D8C2×Q16C2×C8C8C2×C4Q8C4C4C2
# reps11211114222424442

Matrix representation of C89Q16 in GL4(𝔽17) generated by

13000
01300
0001
00130
,
0600
14600
001211
00105
,
81500
7900
0002
0090
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,0,13,0,0,1,0],[0,14,0,0,6,6,0,0,0,0,12,10,0,0,11,5],[8,7,0,0,15,9,0,0,0,0,0,9,0,0,2,0] >;

C89Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_9Q_{16}
% in TeX

G:=Group("C8:9Q16");
// GroupNames label

G:=SmallGroup(128,316);
// by ID

G=gap.SmallGroup(128,316);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,268,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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