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G = C2×C43order 128 = 27

Abelian group of type [2,4,4,4]

direct product, p-group, abelian, monomial

Aliases: C2×C43, SmallGroup(128,997)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C43
C1C2C22C23C24C23×C4C22×C42 — C2×C43
C1 — C2×C43
C1 — C2×C43
C1C23 — C2×C43

Generators and relations for C2×C43
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 636, all normal (4 characteristic)
C1, C2 [×15], C4 [×56], C22 [×35], C2×C4 [×196], C23, C23 [×14], C42 [×112], C22×C4 [×98], C24, C2×C42 [×84], C23×C4 [×7], C43 [×8], C22×C42 [×7], C2×C43
Quotients: C1, C2 [×15], C4 [×56], C22 [×35], C2×C4 [×196], C23 [×15], C42 [×112], C22×C4 [×98], C24, C2×C42 [×84], C23×C4 [×7], C43 [×8], C22×C42 [×7], C2×C43

Smallest permutation representation of C2×C43
Regular action on 128 points
Generators in S128
(1 73)(2 74)(3 75)(4 76)(5 128)(6 125)(7 126)(8 127)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(77 81)(78 82)(79 83)(80 84)(85 89)(86 90)(87 91)(88 92)(93 97)(94 98)(95 99)(96 100)(101 105)(102 106)(103 107)(104 108)(109 113)(110 114)(111 115)(112 116)(117 121)(118 122)(119 123)(120 124)
(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 25 9 85)(2 26 10 86)(3 27 11 87)(4 28 12 88)(5 116 71 54)(6 113 72 55)(7 114 69 56)(8 115 70 53)(13 89 73 29)(14 90 74 30)(15 91 75 31)(16 92 76 32)(17 93 77 33)(18 94 78 34)(19 95 79 35)(20 96 80 36)(21 97 81 37)(22 98 82 38)(23 99 83 39)(24 100 84 40)(41 119 103 57)(42 120 104 58)(43 117 101 59)(44 118 102 60)(45 123 107 61)(46 124 108 62)(47 121 105 63)(48 122 106 64)(49 127 111 66)(50 128 112 67)(51 125 109 68)(52 126 110 65)
(1 109 101 77)(2 110 102 78)(3 111 103 79)(4 112 104 80)(5 124 100 92)(6 121 97 89)(7 122 98 90)(8 123 99 91)(9 51 43 17)(10 52 44 18)(11 49 41 19)(12 50 42 20)(13 55 47 21)(14 56 48 22)(15 53 45 23)(16 54 46 24)(25 68 59 33)(26 65 60 34)(27 66 57 35)(28 67 58 36)(29 72 63 37)(30 69 64 38)(31 70 61 39)(32 71 62 40)(73 113 105 81)(74 114 106 82)(75 115 107 83)(76 116 108 84)(85 125 117 93)(86 126 118 94)(87 127 119 95)(88 128 120 96)

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

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

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

128 conjugacy classes

class 1 2A···2O4A···4DH
order12···24···4
size11···11···1

128 irreducible representations

dim1111
type+++
imageC1C2C2C4
kernelC2×C43C43C22×C42C2×C42
# reps187112

Matrix representation of C2×C43 in GL4(𝔽5) generated by

4000
0400
0010
0001
,
3000
0300
0020
0004
,
1000
0300
0020
0003
,
4000
0100
0010
0002
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2] >;

C2×C43 in GAP, Magma, Sage, TeX

C_2\times C_4^3
% in TeX

G:=Group("C2xC4^3");
// GroupNames label

G:=SmallGroup(128,997);
// by ID

G=gap.SmallGroup(128,997);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,232,352]);
// Polycyclic

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

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