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G = C4⋊C4⋊C8order 128 = 27

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

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

Aliases: C4⋊C4⋊C8, (C2×C4).99D8, C2.4(D4⋊C8), C2.3(Q8⋊C8), (C2×C4).45Q16, C22.29C4≀C2, (C2×C4).82SD16, (C2×C4).7M4(2), (C22×C4).632D4, C2.1(C4.10D8), C22.46(C22⋊C8), C22.31(C23⋊C4), (C2×C42).119C22, C2.2(C22.SD16), C22.36(D4⋊C4), C23.210(C22⋊C4), C22.25(Q8⋊C4), C2.1(C23.31D4), C22.7C42.1C2, C22.13(C4.10D4), C23.65C23.1C2, C2.1(C42.2C22), C2.3(C22.M4(2)), (C2×C4⋊C4).2C4, (C2×C4).7(C2×C8), (C22×C4).143(C2×C4), SmallGroup(128,3)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4⋊C4⋊C8
C1C2C22C23C22×C4C2×C42C23.65C23 — C4⋊C4⋊C8
C1C22C2×C4 — C4⋊C4⋊C8
C1C23C2×C42 — C4⋊C4⋊C8
C1C22C23C2×C42 — C4⋊C4⋊C8

Generators and relations for C4⋊C4⋊C8
 G = < a,b,c | a4=b4=c8=1, bab-1=a-1, cac-1=ab2, cbc-1=ab >

Subgroups: 168 in 79 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×7], C4 [×9], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×13], C23, C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C22×C4 [×3], C22×C4 [×2], C2.C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C22×C8 [×2], C22.7C42 [×2], C23.65C23, C4⋊C4⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), D8, SD16 [×2], Q16, C22⋊C8, C23⋊C4, C4.10D4, D4⋊C4, Q8⋊C4, C4≀C2 [×2], C22.M4(2), D4⋊C8, Q8⋊C8, C22.SD16, C23.31D4, C42.2C22, C4.10D8, C4⋊C4⋊C8

Smallest permutation representation of C4⋊C4⋊C8
Regular action on 128 points
Generators in S128
(1 63 21 115)(2 82 22 30)(3 57 23 117)(4 84 24 32)(5 59 17 119)(6 86 18 26)(7 61 19 113)(8 88 20 28)(9 67 33 41)(10 98 34 128)(11 69 35 43)(12 100 36 122)(13 71 37 45)(14 102 38 124)(15 65 39 47)(16 104 40 126)(25 109 85 53)(27 111 87 55)(29 105 81 49)(31 107 83 51)(42 93 68 76)(44 95 70 78)(46 89 72 80)(48 91 66 74)(50 116 106 64)(52 118 108 58)(54 120 110 60)(56 114 112 62)(73 125 90 103)(75 127 92 97)(77 121 94 99)(79 123 96 101)
(1 99 105 69)(2 95 106 12)(3 45 107 123)(4 38 108 80)(5 103 109 65)(6 91 110 16)(7 41 111 127)(8 34 112 76)(9 27 92 113)(10 56 93 20)(11 63 94 81)(13 31 96 117)(14 52 89 24)(15 59 90 85)(17 125 53 47)(18 74 54 40)(19 67 55 97)(21 121 49 43)(22 78 50 36)(23 71 51 101)(25 39 119 73)(26 66 120 104)(28 128 114 42)(29 35 115 77)(30 70 116 100)(32 124 118 46)(33 87 75 61)(37 83 79 57)(44 64 122 82)(48 60 126 86)(58 72 84 102)(62 68 88 98)
(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)| (1,63,21,115)(2,82,22,30)(3,57,23,117)(4,84,24,32)(5,59,17,119)(6,86,18,26)(7,61,19,113)(8,88,20,28)(9,67,33,41)(10,98,34,128)(11,69,35,43)(12,100,36,122)(13,71,37,45)(14,102,38,124)(15,65,39,47)(16,104,40,126)(25,109,85,53)(27,111,87,55)(29,105,81,49)(31,107,83,51)(42,93,68,76)(44,95,70,78)(46,89,72,80)(48,91,66,74)(50,116,106,64)(52,118,108,58)(54,120,110,60)(56,114,112,62)(73,125,90,103)(75,127,92,97)(77,121,94,99)(79,123,96,101), (1,99,105,69)(2,95,106,12)(3,45,107,123)(4,38,108,80)(5,103,109,65)(6,91,110,16)(7,41,111,127)(8,34,112,76)(9,27,92,113)(10,56,93,20)(11,63,94,81)(13,31,96,117)(14,52,89,24)(15,59,90,85)(17,125,53,47)(18,74,54,40)(19,67,55,97)(21,121,49,43)(22,78,50,36)(23,71,51,101)(25,39,119,73)(26,66,120,104)(28,128,114,42)(29,35,115,77)(30,70,116,100)(32,124,118,46)(33,87,75,61)(37,83,79,57)(44,64,122,82)(48,60,126,86)(58,72,84,102)(62,68,88,98), (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( (1,63,21,115)(2,82,22,30)(3,57,23,117)(4,84,24,32)(5,59,17,119)(6,86,18,26)(7,61,19,113)(8,88,20,28)(9,67,33,41)(10,98,34,128)(11,69,35,43)(12,100,36,122)(13,71,37,45)(14,102,38,124)(15,65,39,47)(16,104,40,126)(25,109,85,53)(27,111,87,55)(29,105,81,49)(31,107,83,51)(42,93,68,76)(44,95,70,78)(46,89,72,80)(48,91,66,74)(50,116,106,64)(52,118,108,58)(54,120,110,60)(56,114,112,62)(73,125,90,103)(75,127,92,97)(77,121,94,99)(79,123,96,101), (1,99,105,69)(2,95,106,12)(3,45,107,123)(4,38,108,80)(5,103,109,65)(6,91,110,16)(7,41,111,127)(8,34,112,76)(9,27,92,113)(10,56,93,20)(11,63,94,81)(13,31,96,117)(14,52,89,24)(15,59,90,85)(17,125,53,47)(18,74,54,40)(19,67,55,97)(21,121,49,43)(22,78,50,36)(23,71,51,101)(25,39,119,73)(26,66,120,104)(28,128,114,42)(29,35,115,77)(30,70,116,100)(32,124,118,46)(33,87,75,61)(37,83,79,57)(44,64,122,82)(48,60,126,86)(58,72,84,102)(62,68,88,98), (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([(1,63,21,115),(2,82,22,30),(3,57,23,117),(4,84,24,32),(5,59,17,119),(6,86,18,26),(7,61,19,113),(8,88,20,28),(9,67,33,41),(10,98,34,128),(11,69,35,43),(12,100,36,122),(13,71,37,45),(14,102,38,124),(15,65,39,47),(16,104,40,126),(25,109,85,53),(27,111,87,55),(29,105,81,49),(31,107,83,51),(42,93,68,76),(44,95,70,78),(46,89,72,80),(48,91,66,74),(50,116,106,64),(52,118,108,58),(54,120,110,60),(56,114,112,62),(73,125,90,103),(75,127,92,97),(77,121,94,99),(79,123,96,101)], [(1,99,105,69),(2,95,106,12),(3,45,107,123),(4,38,108,80),(5,103,109,65),(6,91,110,16),(7,41,111,127),(8,34,112,76),(9,27,92,113),(10,56,93,20),(11,63,94,81),(13,31,96,117),(14,52,89,24),(15,59,90,85),(17,125,53,47),(18,74,54,40),(19,67,55,97),(21,121,49,43),(22,78,50,36),(23,71,51,101),(25,39,119,73),(26,66,120,104),(28,128,114,42),(29,35,115,77),(30,70,116,100),(32,124,118,46),(33,87,75,61),(37,83,79,57),(44,64,122,82),(48,60,126,86),(58,72,84,102),(62,68,88,98)], [(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)])

38 conjugacy classes

class 1 2A···2G4A···4H4I4J4K4L4M4N8A···8P
order12···24···44444448···8
size11···12···24488884···4

38 irreducible representations

dim1111122222244
type+++++-+-
imageC1C2C2C4C8D4M4(2)D8SD16Q16C4≀C2C23⋊C4C4.10D4
kernelC4⋊C4⋊C8C22.7C42C23.65C23C2×C4⋊C4C4⋊C4C22×C4C2×C4C2×C4C2×C4C2×C4C22C22C22
# reps1214822242811

Matrix representation of C4⋊C4⋊C8 in GL6(𝔽17)

1600000
0160000
004400
0001300
0000016
000010
,
7160000
14100000
00141000
006300
00001016
0000167
,
14150000
1530000
0041000
000100
00001113
0000136

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[7,14,0,0,0,0,16,10,0,0,0,0,0,0,14,6,0,0,0,0,10,3,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[14,15,0,0,0,0,15,3,0,0,0,0,0,0,4,0,0,0,0,0,10,1,0,0,0,0,0,0,11,13,0,0,0,0,13,6] >;

C4⋊C4⋊C8 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes C_8
% in TeX

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

G:=SmallGroup(128,3);
// by ID

G=gap.SmallGroup(128,3);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,2,56,85,120,422,387,184,794,248]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^8=1,b*a*b^-1=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=a*b>;
// generators/relations

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