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G = C429C8order 128 = 27

6th semidirect product of C42 and C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C429C8, C43.9C2, C42.57Q8, C42.323D4, C41(C4⋊C8), C4.44(C4⋊Q8), (C2×C42).45C4, C4.44(C41D4), (C2×C4).78M4(2), C2.2(C429C4), (C22×C8).23C22, C22.38(C22×C8), C2.5(C4⋊M4(2)), C23.267(C22×C4), C22.49(C2×M4(2)), (C22×C4).1625C23, (C2×C42).1058C22, C2.10(C2×C4⋊C8), (C2×C4⋊C8).26C2, (C2×C4).85(C2×C8), (C2×C4).88(C4⋊C4), C22.65(C2×C4⋊C4), (C2×C4).339(C2×Q8), (C2×C4).1523(C2×D4), (C22×C4).483(C2×C4), SmallGroup(128,574)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C429C8
C1C2C4C2×C4C22×C4C2×C42C43 — C429C8
C1C22 — C429C8
C1C22×C4 — C429C8
C1C2C2C22×C4 — C429C8

Generators and relations for C429C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 204 in 150 conjugacy classes, 104 normal (9 characteristic)
C1, C2, C2 [×6], C4, C4 [×15], C4 [×6], C22, C22 [×6], C8 [×4], C2×C4 [×30], C2×C4 [×6], C23, C42 [×16], C42 [×6], C2×C8 [×12], C22×C4, C22×C4 [×6], C4⋊C8 [×12], C2×C42, C2×C42 [×6], C22×C8 [×4], C43, C2×C4⋊C8 [×6], C429C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C2×C8 [×6], M4(2) [×6], C22×C4, C2×D4 [×3], C2×Q8 [×3], C4⋊C8 [×12], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C22×C8, C2×M4(2) [×3], C429C4, C2×C4⋊C8 [×3], C4⋊M4(2) [×3], C429C8

Smallest permutation representation of C429C8
Regular action on 128 points
Generators in S128
(1 35 23 91)(2 92 24 36)(3 37 17 93)(4 94 18 38)(5 39 19 95)(6 96 20 40)(7 33 21 89)(8 90 22 34)(9 86 74 30)(10 31 75 87)(11 88 76 32)(12 25 77 81)(13 82 78 26)(14 27 79 83)(15 84 80 28)(16 29 73 85)(41 117 97 61)(42 62 98 118)(43 119 99 63)(44 64 100 120)(45 113 101 57)(46 58 102 114)(47 115 103 59)(48 60 104 116)(49 125 105 65)(50 66 106 126)(51 127 107 67)(52 68 108 128)(53 121 109 69)(54 70 110 122)(55 123 111 71)(56 72 112 124)
(1 107 103 83)(2 84 104 108)(3 109 97 85)(4 86 98 110)(5 111 99 87)(6 88 100 112)(7 105 101 81)(8 82 102 106)(9 62 70 38)(10 39 71 63)(11 64 72 40)(12 33 65 57)(13 58 66 34)(14 35 67 59)(15 60 68 36)(16 37 69 61)(17 53 41 29)(18 30 42 54)(19 55 43 31)(20 32 44 56)(21 49 45 25)(22 26 46 50)(23 51 47 27)(24 28 48 52)(73 93 121 117)(74 118 122 94)(75 95 123 119)(76 120 124 96)(77 89 125 113)(78 114 126 90)(79 91 127 115)(80 116 128 92)
(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)| (1,35,23,91)(2,92,24,36)(3,37,17,93)(4,94,18,38)(5,39,19,95)(6,96,20,40)(7,33,21,89)(8,90,22,34)(9,86,74,30)(10,31,75,87)(11,88,76,32)(12,25,77,81)(13,82,78,26)(14,27,79,83)(15,84,80,28)(16,29,73,85)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,65)(50,66,106,126)(51,127,107,67)(52,68,108,128)(53,121,109,69)(54,70,110,122)(55,123,111,71)(56,72,112,124), (1,107,103,83)(2,84,104,108)(3,109,97,85)(4,86,98,110)(5,111,99,87)(6,88,100,112)(7,105,101,81)(8,82,102,106)(9,62,70,38)(10,39,71,63)(11,64,72,40)(12,33,65,57)(13,58,66,34)(14,35,67,59)(15,60,68,36)(16,37,69,61)(17,53,41,29)(18,30,42,54)(19,55,43,31)(20,32,44,56)(21,49,45,25)(22,26,46,50)(23,51,47,27)(24,28,48,52)(73,93,121,117)(74,118,122,94)(75,95,123,119)(76,120,124,96)(77,89,125,113)(78,114,126,90)(79,91,127,115)(80,116,128,92), (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( (1,35,23,91)(2,92,24,36)(3,37,17,93)(4,94,18,38)(5,39,19,95)(6,96,20,40)(7,33,21,89)(8,90,22,34)(9,86,74,30)(10,31,75,87)(11,88,76,32)(12,25,77,81)(13,82,78,26)(14,27,79,83)(15,84,80,28)(16,29,73,85)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,65)(50,66,106,126)(51,127,107,67)(52,68,108,128)(53,121,109,69)(54,70,110,122)(55,123,111,71)(56,72,112,124), (1,107,103,83)(2,84,104,108)(3,109,97,85)(4,86,98,110)(5,111,99,87)(6,88,100,112)(7,105,101,81)(8,82,102,106)(9,62,70,38)(10,39,71,63)(11,64,72,40)(12,33,65,57)(13,58,66,34)(14,35,67,59)(15,60,68,36)(16,37,69,61)(17,53,41,29)(18,30,42,54)(19,55,43,31)(20,32,44,56)(21,49,45,25)(22,26,46,50)(23,51,47,27)(24,28,48,52)(73,93,121,117)(74,118,122,94)(75,95,123,119)(76,120,124,96)(77,89,125,113)(78,114,126,90)(79,91,127,115)(80,116,128,92), (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([(1,35,23,91),(2,92,24,36),(3,37,17,93),(4,94,18,38),(5,39,19,95),(6,96,20,40),(7,33,21,89),(8,90,22,34),(9,86,74,30),(10,31,75,87),(11,88,76,32),(12,25,77,81),(13,82,78,26),(14,27,79,83),(15,84,80,28),(16,29,73,85),(41,117,97,61),(42,62,98,118),(43,119,99,63),(44,64,100,120),(45,113,101,57),(46,58,102,114),(47,115,103,59),(48,60,104,116),(49,125,105,65),(50,66,106,126),(51,127,107,67),(52,68,108,128),(53,121,109,69),(54,70,110,122),(55,123,111,71),(56,72,112,124)], [(1,107,103,83),(2,84,104,108),(3,109,97,85),(4,86,98,110),(5,111,99,87),(6,88,100,112),(7,105,101,81),(8,82,102,106),(9,62,70,38),(10,39,71,63),(11,64,72,40),(12,33,65,57),(13,58,66,34),(14,35,67,59),(15,60,68,36),(16,37,69,61),(17,53,41,29),(18,30,42,54),(19,55,43,31),(20,32,44,56),(21,49,45,25),(22,26,46,50),(23,51,47,27),(24,28,48,52),(73,93,121,117),(74,118,122,94),(75,95,123,119),(76,120,124,96),(77,89,125,113),(78,114,126,90),(79,91,127,115),(80,116,128,92)], [(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)])

56 conjugacy classes

class 1 2A···2G4A···4H4I···4AF8A···8P
order12···24···44···48···8
size11···11···12···24···4

56 irreducible representations

dim11111222
type++++-
imageC1C2C2C4C8D4Q8M4(2)
kernelC429C8C43C2×C4⋊C8C2×C42C42C42C42C2×C4
# reps1168166612

Matrix representation of C429C8 in GL5(𝔽17)

10000
04000
001300
00042
000013
,
10000
04000
001300
00010
00001
,
90000
00100
016000
00058
0001412

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

C429C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9C_8
% in TeX

G:=Group("C4^2:9C8");
// GroupNames label

G:=SmallGroup(128,574);
// by ID

G=gap.SmallGroup(128,574);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,100,124]);
// Polycyclic

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

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