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## G = C4×C4⋊C8order 128 = 27

### Direct product of C4 and C4⋊C8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C4⋊C8
G = < a,b,c | a4=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 196 in 158 conjugacy classes, 120 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C4×C8, C2×C4⋊C8, C4×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×M4(2), C4×C4⋊C4, C2×C4×C8, C4×M4(2), C2×C4⋊C8, C42.12C4, C4×C4⋊C8

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

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 1 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C8 D4 Q8 M4(2) C4○D4 kernel C4×C4⋊C8 C22.7C42 C43 C2×C4×C8 C2×C4⋊C8 C4⋊C8 C2×C42 C42 C42 C42 C2×C4 C2×C4 # reps 1 2 1 2 2 16 8 32 2 2 8 4

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

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

C4×C4⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes C_8
% in TeX

G:=Group("C4xC4:C8");
// GroupNames label

G:=SmallGroup(128,498);
// by ID

G=gap.SmallGroup(128,498);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,124]);
// Polycyclic

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

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