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## G = C2×C42.2C22order 128 = 27

### Direct product of C2 and C42.2C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.2C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C42.C2 — C2×C42.2C22
 Lower central C1 — C22 — C2×C4 — C2×C42.2C22
 Upper central C1 — C23 — C2×C42 — C2×C42.2C22
 Jennings C1 — C22 — C22 — C42 — C2×C42.2C22

Generators and relations for C2×C42.2C22
G = < a,b,c,d,e | a2=b4=c4=1, d2=c, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe-1=b-1, cd=dc, ece-1=b2c-1, ede-1=b-1c2d >

Subgroups: 196 in 116 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×10], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], C23, C42 [×2], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C42.2C22 [×4], C2×C8⋊C4 [×2], C2×C42.C2, C2×C42.2C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C42.2C22 [×4], C2×C4.10D4, C2×C4≀C2 [×2], C2×C42.2C22

Smallest permutation representation of C2×C42.2C22
Regular action on 128 points
Generators in S128
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 109)(26 110)(27 111)(28 112)(29 105)(30 106)(31 107)(32 108)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(49 125)(50 126)(51 127)(52 128)(53 121)(54 122)(55 123)(56 124)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(73 84)(74 85)(75 86)(76 87)(77 88)(78 81)(79 82)(80 83)
(1 39 111 95)(2 36 112 92)(3 33 105 89)(4 38 106 94)(5 35 107 91)(6 40 108 96)(7 37 109 93)(8 34 110 90)(9 17 121 88)(10 22 122 85)(11 19 123 82)(12 24 124 87)(13 21 125 84)(14 18 126 81)(15 23 127 86)(16 20 128 83)(25 57 97 72)(26 62 98 69)(27 59 99 66)(28 64 100 71)(29 61 101 68)(30 58 102 65)(31 63 103 70)(32 60 104 67)(41 49 73 113)(42 54 74 118)(43 51 75 115)(44 56 76 120)(45 53 77 117)(46 50 78 114)(47 55 79 119)(48 52 80 116)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(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)
(1 119 5 115)(2 76 6 80)(3 53 7 49)(4 42 8 46)(9 25 13 29)(10 69 14 65)(11 103 15 99)(12 60 16 64)(17 72 21 68)(18 102 22 98)(19 63 23 59)(20 28 24 32)(26 81 30 85)(27 123 31 127)(33 45 37 41)(34 114 38 118)(35 75 39 79)(36 56 40 52)(43 95 47 91)(44 108 48 112)(50 94 54 90)(51 111 55 107)(57 84 61 88)(58 122 62 126)(66 82 70 86)(67 128 71 124)(73 89 77 93)(74 110 78 106)(83 100 87 104)(92 120 96 116)(97 125 101 121)(105 117 109 113)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 D4 D4 C4≀C2 C4.10D4 kernel C2×C42.2C22 C42.2C22 C2×C8⋊C4 C2×C42.C2 C2×C4⋊C4 C42.C2 C42 C22×C4 C22 C22 # reps 1 4 2 1 4 4 2 2 16 2

Matrix representation of C2×C42.2C22 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 9 0 0 0 0 13 1
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 2 0 0 0 0 1 4
,
 0 13 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 10 12
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 11 10 0 0 0 0 5 6

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,1,0,0,0,0,2,4],[0,1,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,10,0,0,0,0,3,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,11,5,0,0,0,0,10,6] >;

C2×C42.2C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._2C_2^2
% in TeX

G:=Group("C2xC4^2.2C2^2");
// GroupNames label

G:=SmallGroup(128,255);
// by ID

G=gap.SmallGroup(128,255);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971,102]);
// Polycyclic

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

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