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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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22 [×3], C22 [×4], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C16 [×2], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×4], C22×C4, C22×C4 [×2], C2.D8 [×4], C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], C2×C4⋊C4 [×2], C22×C8, C2×C2.D8 [×2], C22×C16, C8.7C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, D16, SD32 [×2], Q32, C22.4Q16, C2.D16 [×2], C2.Q32 [×2], C163C4, C164C4, C8.7C42

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