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G = C4⋊C813C4order 128 = 27

9th semidirect product of C4⋊C8 and C4 acting via C4/C2=C2

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

Aliases: C4⋊C813C4, C88(C4⋊C4), (C4×C8)⋊24C4, C42(C8⋊C4), (C2×C8).58Q8, C4.45(C4×Q8), (C2×C8).287D4, C4.169(C4×D4), C2.2(C86D4), (C2×C4).72C42, C2.1(C84Q8), C22.91(C4×D4), C22.23(C4×Q8), C42.314(C2×C4), (C2×C4).45M4(2), C22.27(C8○D4), C2.C42.9C4, C22.55(C2×C42), C4.74(C42⋊C2), (C22×C8).384C22, C23.260(C22×C4), C22.42(C2×M4(2)), C2.16(C82M4(2)), (C22×C4).1613C23, (C2×C42).1049C22, C22.7C42.42C2, C2.9(C4×C4⋊C4), (C2×C4×C8).62C2, (C4×C4⋊C4).7C2, C4.74(C2×C4⋊C4), (C2×C4⋊C4).48C4, (C2×C4⋊C8).53C2, C2.10(C2×C8⋊C4), (C2×C8).137(C2×C4), (C2×C4).333(C2×Q8), (C2×C8⋊C4).26C2, (C2×C4).1505(C2×D4), (C2×C4).923(C4○D4), (C22×C4).110(C2×C4), (C2×C4).603(C22×C4), SmallGroup(128,502)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊C813C4
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — C4⋊C813C4
C1C22 — C4⋊C813C4
C1C22×C4 — C4⋊C813C4
C1C2C2C22×C4 — C4⋊C813C4

Generators and relations for C4⋊C813C4
 G = < a,b,c | a4=b8=c4=1, bab-1=cac-1=a-1, cbc-1=b5 >

Subgroups: 188 in 136 conjugacy classes, 92 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×4], C8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4 [×2], C2×C4⋊C8, C4⋊C813C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], M4(2) [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C8⋊C4 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C8○D4 [×2], C4×C4⋊C4, C2×C8⋊C4, C82M4(2), C86D4 [×2], C84Q8 [×2], C4⋊C813C4

Smallest permutation representation of C4⋊C813C4
Regular action on 128 points
Generators in S128
(1 70 29 113)(2 114 30 71)(3 72 31 115)(4 116 32 65)(5 66 25 117)(6 118 26 67)(7 68 27 119)(8 120 28 69)(9 38 59 22)(10 23 60 39)(11 40 61 24)(12 17 62 33)(13 34 63 18)(14 19 64 35)(15 36 57 20)(16 21 58 37)(41 75 85 108)(42 109 86 76)(43 77 87 110)(44 111 88 78)(45 79 81 112)(46 105 82 80)(47 73 83 106)(48 107 84 74)(49 104 121 93)(50 94 122 97)(51 98 123 95)(52 96 124 99)(53 100 125 89)(54 90 126 101)(55 102 127 91)(56 92 128 103)
(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 18 93 82)(2 23 94 87)(3 20 95 84)(4 17 96 81)(5 22 89 86)(6 19 90 83)(7 24 91 88)(8 21 92 85)(9 125 76 117)(10 122 77 114)(11 127 78 119)(12 124 79 116)(13 121 80 113)(14 126 73 118)(15 123 74 115)(16 128 75 120)(25 38 100 42)(26 35 101 47)(27 40 102 44)(28 37 103 41)(29 34 104 46)(30 39 97 43)(31 36 98 48)(32 33 99 45)(49 105 70 63)(50 110 71 60)(51 107 72 57)(52 112 65 62)(53 109 66 59)(54 106 67 64)(55 111 68 61)(56 108 69 58)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim111111111122222
type+++++++-
imageC1C2C2C2C2C2C4C4C4C4D4Q8M4(2)C4○D4C8○D4
kernelC4⋊C813C4C22.7C42C4×C4⋊C4C2×C4×C8C2×C8⋊C4C2×C4⋊C8C2.C42C4×C8C4⋊C8C2×C4⋊C4C2×C8C2×C8C2×C4C2×C4C22
# reps121121488422848

Matrix representation of C4⋊C813C4 in GL5(𝔽17)

10000
04000
001300
000160
000016
,
130000
00200
02000
000015
000150
,
130000
00100
016000
00001
000160

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

C4⋊C813C4 in GAP, Magma, Sage, TeX

C_4\rtimes C_8\rtimes_{13}C_4
% in TeX

G:=Group("C4:C8:13C4");
// GroupNames label

G:=SmallGroup(128,502);
// by ID

G=gap.SmallGroup(128,502);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,142,172]);
// Polycyclic

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

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