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G = C8⋊C42order 128 = 27

The semidirect product of C8 and C42 acting via C42/C22=C22

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

Aliases: C8⋊C42, C42.22Q8, C4.Q86C4, C8⋊C47C4, C4.5(C4×Q8), C2.D811C4, C4.25(C2×C42), C22.97(C4×D4), C2.3(D8⋊C4), C42.134(C2×C4), C23.736(C2×D4), (C22×C4).673D4, C2.3(Q16⋊C4), C4.49(C42⋊C2), C2.4(SD16⋊C4), C22.56(C8⋊C22), C22.4Q16.46C2, (C2×C42).241C22, (C22×C8).210C22, C2.3(M4(2)⋊C4), (C22×C4).1317C23, C22.45(C8.C22), C2.15(C4×C4⋊C4), (C4×C4⋊C4).11C2, (C2×C8⋊C4).2C2, (C2×C4.Q8).3C2, C4⋊C4.148(C2×C4), (C2×C4).80(C4⋊C4), (C2×C8).140(C2×C4), C22.60(C2×C4⋊C4), (C2×C4).186(C2×Q8), (C2×C2.D8).32C2, (C2×C4).548(C4○D4), (C2×C4⋊C4).752C22, (C2×C4).524(C22×C4), SmallGroup(128,508)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8⋊C42
C1C2C22C2×C4C22×C4C2×C42C4×C4⋊C4 — C8⋊C42
C1C2C4 — C8⋊C42
C1C23C2×C42 — C8⋊C42
C1C2C2C22×C4 — C8⋊C42

Generators and relations for C8⋊C42
 G = < a,b,c | a8=b4=c4=1, bab-1=a-1, cac-1=a5, bc=cb >

Subgroups: 228 in 136 conjugacy classes, 84 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×20], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×2], C4×C4⋊C4 [×2], C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C8⋊C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C8⋊C22 [×2], C8.C22 [×2], C4×C4⋊C4, M4(2)⋊C4 [×2], SD16⋊C4 [×2], Q16⋊C4, D8⋊C4, C8⋊C42

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

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4AB8A···8H
order12···24···44···48···8
size11···12···24···44···4

44 irreducible representations

dim11111111122244
type++++++-++-
imageC1C2C2C2C2C2C4C4C4Q8D4C4○D4C8⋊C22C8.C22
kernelC8⋊C42C22.4Q16C4×C4⋊C4C2×C8⋊C4C2×C4.Q8C2×C2.D8C8⋊C4C4.Q8C2.D8C42C22×C4C2×C4C22C22
# reps12211188822422

Matrix representation of C8⋊C42 in GL8(𝔽17)

216000000
515000000
00620000
007110000
00002151515
000022215
000022152
00001521515
,
130000000
14000000
001150000
001160000
00001211612
000015121
00001612121
000012115
,
160000000
016000000
00400000
00040000
00000010
00000001
00001000
00000100

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

C8⋊C42 in GAP, Magma, Sage, TeX

C_8\rtimes C_4^2
% in TeX

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

G:=SmallGroup(128,508);
// by ID

G=gap.SmallGroup(128,508);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,1018,248,1411,172]);
// Polycyclic

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

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