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G = C2×C4×C16order 128 = 27

Abelian group of type [2,4,16]

direct product, p-group, abelian, monomial

Aliases: C2×C4×C16, SmallGroup(128,837)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C4×C16
C1C2C4C2×C4C2×C8C22×C8C2×C4×C8 — C2×C4×C16
C1 — C2×C4×C16
C1 — C2×C4×C16
C1C2C2C2C2C4C4C2×C8 — C2×C4×C16

Generators and relations for C2×C4×C16
 G = < a,b,c | a2=b4=c16=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (16 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×10], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C16 [×8], C42 [×4], C2×C8 [×2], C2×C8 [×10], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C16 [×12], C2×C42, C22×C8 [×2], C4×C16 [×4], C2×C4×C8, C22×C16 [×2], C2×C4×C16
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C16 [×8], C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], C2×C16 [×12], C2×C42, C22×C8 [×2], C4×C16 [×4], C2×C4×C8, C22×C16 [×2], C2×C4×C16

Smallest permutation representation of C2×C4×C16
Regular action on 128 points
Generators in S128
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 125)(34 126)(35 127)(36 128)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 81)(63 82)(64 83)
(1 20 117 86)(2 21 118 87)(3 22 119 88)(4 23 120 89)(5 24 121 90)(6 25 122 91)(7 26 123 92)(8 27 124 93)(9 28 125 94)(10 29 126 95)(11 30 127 96)(12 31 128 81)(13 32 113 82)(14 17 114 83)(15 18 115 84)(16 19 116 85)(33 59 65 100)(34 60 66 101)(35 61 67 102)(36 62 68 103)(37 63 69 104)(38 64 70 105)(39 49 71 106)(40 50 72 107)(41 51 73 108)(42 52 74 109)(43 53 75 110)(44 54 76 111)(45 55 77 112)(46 56 78 97)(47 57 79 98)(48 58 80 99)
(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)| (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,81)(63,82)(64,83), (1,20,117,86)(2,21,118,87)(3,22,119,88)(4,23,120,89)(5,24,121,90)(6,25,122,91)(7,26,123,92)(8,27,124,93)(9,28,125,94)(10,29,126,95)(11,30,127,96)(12,31,128,81)(13,32,113,82)(14,17,114,83)(15,18,115,84)(16,19,116,85)(33,59,65,100)(34,60,66,101)(35,61,67,102)(36,62,68,103)(37,63,69,104)(38,64,70,105)(39,49,71,106)(40,50,72,107)(41,51,73,108)(42,52,74,109)(43,53,75,110)(44,54,76,111)(45,55,77,112)(46,56,78,97)(47,57,79,98)(48,58,80,99), (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( (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,81)(63,82)(64,83), (1,20,117,86)(2,21,118,87)(3,22,119,88)(4,23,120,89)(5,24,121,90)(6,25,122,91)(7,26,123,92)(8,27,124,93)(9,28,125,94)(10,29,126,95)(11,30,127,96)(12,31,128,81)(13,32,113,82)(14,17,114,83)(15,18,115,84)(16,19,116,85)(33,59,65,100)(34,60,66,101)(35,61,67,102)(36,62,68,103)(37,63,69,104)(38,64,70,105)(39,49,71,106)(40,50,72,107)(41,51,73,108)(42,52,74,109)(43,53,75,110)(44,54,76,111)(45,55,77,112)(46,56,78,97)(47,57,79,98)(48,58,80,99), (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([(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,125),(34,126),(35,127),(36,128),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,96),(62,81),(63,82),(64,83)], [(1,20,117,86),(2,21,118,87),(3,22,119,88),(4,23,120,89),(5,24,121,90),(6,25,122,91),(7,26,123,92),(8,27,124,93),(9,28,125,94),(10,29,126,95),(11,30,127,96),(12,31,128,81),(13,32,113,82),(14,17,114,83),(15,18,115,84),(16,19,116,85),(33,59,65,100),(34,60,66,101),(35,61,67,102),(36,62,68,103),(37,63,69,104),(38,64,70,105),(39,49,71,106),(40,50,72,107),(41,51,73,108),(42,52,74,109),(43,53,75,110),(44,54,76,111),(45,55,77,112),(46,56,78,97),(47,57,79,98),(48,58,80,99)], [(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···2G4A···4X8A···8AF16A···16BL
order12···24···48···816···16
size11···11···11···11···1

128 irreducible representations

dim111111111111
type++++
imageC1C2C2C2C4C4C4C4C8C8C8C16
kernelC2×C4×C16C4×C16C2×C4×C8C22×C16C4×C8C2×C16C2×C42C22×C8C42C2×C8C22×C4C2×C4
# reps141241622816864

Matrix representation of C2×C4×C16 in GL3(𝔽17) generated by

1600
010
001
,
1600
0130
0016
,
400
0160
007
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,13,0,0,0,16],[4,0,0,0,16,0,0,0,7] >;

C2×C4×C16 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{16}
% in TeX

G:=Group("C2xC4xC16");
// GroupNames label

G:=SmallGroup(128,837);
// by ID

G=gap.SmallGroup(128,837);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,136,124]);
// Polycyclic

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

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