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G = C8.7C42order 128 = 27

1st non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.7C42, C22.7D16, C23.55D8, C22.4Q32, C22.9SD32, C2.D83C4, (C2×C16)⋊11C4, C8.12(C4⋊C4), (C2×C8).41Q8, (C2×C8).337D4, (C2×C4).31Q16, C4.8(C4.Q8), (C22×C16).3C2, (C2×C4).64SD16, C2.2(C163C4), C2.2(C164C4), C2.2(C2.D16), C8.33(C22⋊C4), (C22×C4).569D4, C4.13(Q8⋊C4), C2.2(C2.Q32), C22.18(C2.D8), C4.1(C2.C42), (C22×C8).521C22, C22.44(D4⋊C4), C2.10(C22.4Q16), (C2×C2.D8).1C2, (C2×C8).167(C2×C4), (C2×C4).108(C4⋊C4), (C2×C4).227(C22⋊C4), SmallGroup(128,112)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.7C42
C1C2C4C8C2×C8C22×C8C22×C16 — C8.7C42
C1C2C4C8 — C8.7C42
C1C23C22×C4C22×C8 — C8.7C42
C1C2C2C2C2C4C4C22×C8 — C8.7C42

Generators and relations for C8.7C42
 G = < a,b,c | a8=b4=1, c4=a6, bab-1=a-1, ac=ca, cbc-1=a-1b >

Subgroups: 168 in 76 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.D8, C2.D8, C2×C16, C2×C16, C2×C4⋊C4, C22×C8, C2×C2.D8, C22×C16, C8.7C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, D16, SD32, Q32, C22.4Q16, C2.D16, C2.Q32, C163C4, C164C4, C8.7C42

Smallest permutation representation of C8.7C42
Regular action on 128 points
Generators in S128
(1 86 13 82 9 94 5 90)(2 87 14 83 10 95 6 91)(3 88 15 84 11 96 7 92)(4 89 16 85 12 81 8 93)(17 73 29 69 25 65 21 77)(18 74 30 70 26 66 22 78)(19 75 31 71 27 67 23 79)(20 76 32 72 28 68 24 80)(33 53 45 49 41 61 37 57)(34 54 46 50 42 62 38 58)(35 55 47 51 43 63 39 59)(36 56 48 52 44 64 40 60)(97 122 109 118 105 114 101 126)(98 123 110 119 106 115 102 127)(99 124 111 120 107 116 103 128)(100 125 112 121 108 117 104 113)
(1 101 70 59)(2 127 71 36)(3 99 72 57)(4 125 73 34)(5 97 74 55)(6 123 75 48)(7 111 76 53)(8 121 77 46)(9 109 78 51)(10 119 79 44)(11 107 80 49)(12 117 65 42)(13 105 66 63)(14 115 67 40)(15 103 68 61)(16 113 69 38)(17 54 93 112)(18 47 94 122)(19 52 95 110)(20 45 96 120)(21 50 81 108)(22 43 82 118)(23 64 83 106)(24 41 84 116)(25 62 85 104)(26 39 86 114)(27 60 87 102)(28 37 88 128)(29 58 89 100)(30 35 90 126)(31 56 91 98)(32 33 92 124)
(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)

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

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

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

44 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L8A···8H16A···16P
order12···244444···48···816···16
size11···122228···82···22···2

44 irreducible representations

dim11111222222222
type++++-+-++-
imageC1C2C2C4C4D4Q8D4SD16Q16D8D16SD32Q32
kernelC8.7C42C2×C2.D8C22×C16C2.D8C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C23C22C22C22
# reps12184211422484

Matrix representation of C8.7C42 in GL5(𝔽17)

10000
016000
001600
000011
000311
,
40000
011400
0121600
000815
00079
,
130000
011400
012600
000712
000112

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,3,0,0,0,11,11],[4,0,0,0,0,0,1,12,0,0,0,14,16,0,0,0,0,0,8,7,0,0,0,15,9],[13,0,0,0,0,0,11,12,0,0,0,4,6,0,0,0,0,0,7,11,0,0,0,12,2] >;

C8.7C42 in GAP, Magma, Sage, TeX

C_8._7C_4^2
% in TeX

G:=Group("C8.7C4^2");
// GroupNames label

G:=SmallGroup(128,112);
// by ID

G=gap.SmallGroup(128,112);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,3924,102,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations

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