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## G = C42×C8order 128 = 27

### Abelian group of type [4,4,8]

Aliases: C42×C8, SmallGroup(128,456)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42×C8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C42×C8
 Lower central C1 — C42×C8
 Upper central C1 — C42×C8
 Jennings C1 — C2 — C2 — C22×C4 — C42×C8

Generators and relations for C42×C8
G = < a,b,c | a4=b4=c8=1, ab=ba, ac=ca, bc=cb >

Subgroups: 204, all normal (6 characteristic)
C1, C2, C2 [×6], C4 [×28], C22 [×7], C8 [×16], C2×C4 [×42], C23, C42 [×28], C2×C8 [×24], C22×C4, C22×C4 [×6], C4×C8 [×24], C2×C42 [×7], C22×C8 [×4], C43, C2×C4×C8 [×6], C42×C8
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C8 [×16], C2×C4 [×42], C23, C42 [×28], C2×C8 [×24], C22×C4 [×7], C4×C8 [×24], C2×C42 [×7], C22×C8 [×4], C43, C2×C4×C8 [×6], C42×C8

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

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4BD 8A ··· 8BL order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 1 ··· 1

128 irreducible representations

 dim 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C8 kernel C42×C8 C43 C2×C4×C8 C4×C8 C2×C42 C42 # reps 1 1 6 48 8 64

Matrix representation of C42×C8 in GL3(𝔽17) generated by

 4 0 0 0 16 0 0 0 13
,
 1 0 0 0 13 0 0 0 16
,
 16 0 0 0 1 0 0 0 2
G:=sub<GL(3,GF(17))| [4,0,0,0,16,0,0,0,13],[1,0,0,0,13,0,0,0,16],[16,0,0,0,1,0,0,0,2] >;

C42×C8 in GAP, Magma, Sage, TeX

C_4^2\times C_8
% in TeX

G:=Group("C4^2xC8");
// GroupNames label

G:=SmallGroup(128,456);
// by ID

G=gap.SmallGroup(128,456);
# by ID

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

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

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