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## G = C4×C8⋊C4order 128 = 27

### Direct product of C4 and C8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C8⋊C4
G = < a,b,c | a4=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 204 in 180 conjugacy classes, 156 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C4×C8, C8⋊C4, C2×C42, C2×C42, C22×C8, C43, C2×C4×C8, C2×C8⋊C4, C4×C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C43, C2×C8⋊C4, C4×M4(2), C4×C8⋊C4

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

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4AN 8A ··· 8AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C4 M4(2) kernel C4×C8⋊C4 C43 C2×C4×C8 C2×C8⋊C4 C4×C8 C8⋊C4 C2×C42 C2×C4 # reps 1 1 2 4 16 32 8 16

Matrix representation of C4×C8⋊C4 in GL4(𝔽17) generated by

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

C4×C8⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_8\rtimes C_4
% in TeX

G:=Group("C4xC8:C4");
// GroupNames label

G:=SmallGroup(128,457);
// by ID

G=gap.SmallGroup(128,457);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,925,120,184,248]);
// Polycyclic

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

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