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## G = C2×C16⋊4C4order 128 = 27

### Direct product of C2 and C16⋊4C4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C16⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×C16 — C2×C16⋊4C4
 Lower central C1 — C2 — C4 — C8 — C2×C16⋊4C4
 Upper central C1 — C23 — C22×C4 — C22×C8 — C2×C16⋊4C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C16⋊4C4

Generators and relations for C2×C164C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b7 >

Subgroups: 172 in 84 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.D8, C2.D8, C2×C16, C2×C4⋊C4, C22×C8, C164C4, C2×C2.D8, C22×C16, C2×C164C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, SD32, C2×C4⋊C4, C2×D8, C2×Q16, C164C4, C2×C2.D8, C2×SD32, C2×C164C4

Smallest permutation representation of C2×C164C4
Regular action on 128 points
Generators in S128
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 81)(13 82)(14 83)(15 84)(16 85)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 113)(33 80)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 97)(64 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)
(1 112 26 75)(2 103 27 66)(3 110 28 73)(4 101 29 80)(5 108 30 71)(6 99 31 78)(7 106 32 69)(8 97 17 76)(9 104 18 67)(10 111 19 74)(11 102 20 65)(12 109 21 72)(13 100 22 79)(14 107 23 70)(15 98 24 77)(16 105 25 68)(33 89 51 126)(34 96 52 117)(35 87 53 124)(36 94 54 115)(37 85 55 122)(38 92 56 113)(39 83 57 120)(40 90 58 127)(41 81 59 118)(42 88 60 125)(43 95 61 116)(44 86 62 123)(45 93 63 114)(46 84 64 121)(47 91 49 128)(48 82 50 119)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 8A ··· 8H 16A ··· 16P order 1 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 16 ··· 16 size 1 1 ··· 1 2 2 2 2 8 ··· 8 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + + - + image C1 C2 C2 C2 C4 D4 Q8 D4 D8 Q16 D8 SD32 kernel C2×C16⋊4C4 C16⋊4C4 C2×C2.D8 C22×C16 C2×C16 C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 8 1 2 1 2 4 2 16

Matrix representation of C2×C164C4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 16 0 0 0 0 14 0 0 0 0 6
,
 4 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,14,0,0,0,0,6],[4,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0] >;

C2×C164C4 in GAP, Magma, Sage, TeX

C_2\times C_{16}\rtimes_4C_4
% in TeX

G:=Group("C2xC16:4C4");
// GroupNames label

G:=SmallGroup(128,889);
// by ID

G=gap.SmallGroup(128,889);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,736,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations

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