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G = C23×C42order 128 = 27

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

direct product, p-group, abelian, monomial

Aliases: C23×C42, SmallGroup(128,2150)

Series: Derived Chief Lower central Upper central Jennings

C1 — C23×C42
C1C2C22C23C24C25C24×C4 — C23×C42
C1 — C23×C42
C1 — C23×C42
C1C22 — C23×C42

Generators and relations for C23×C42
 G = < a,b,c,d,e | a2=b2=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 1500, all normal (4 characteristic)
C1, C2 [×31], C4 [×48], C22, C22 [×154], C2×C4 [×360], C23 [×155], C42 [×64], C22×C4 [×420], C24 [×31], C2×C42 [×112], C23×C4 [×90], C25, C22×C42 [×28], C24×C4 [×3], C23×C42
Quotients: C1, C2 [×31], C4 [×48], C22 [×155], C2×C4 [×360], C23 [×155], C42 [×64], C22×C4 [×420], C24 [×31], C2×C42 [×112], C23×C4 [×90], C25, C22×C42 [×28], C24×C4 [×3], C23×C42

Smallest permutation representation of C23×C42
Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 13)(4 14)(5 89)(6 90)(7 91)(8 92)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(53 61)(54 62)(55 63)(56 64)(57 68)(58 65)(59 66)(60 67)(69 77)(70 78)(71 79)(72 80)(73 81)(74 82)(75 83)(76 84)(85 93)(86 94)(87 95)(88 96)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
(1 11)(2 12)(3 9)(4 10)(5 93)(6 94)(7 95)(8 96)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 33)(30 34)(31 35)(32 36)(37 41)(38 42)(39 43)(40 44)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 60)(61 68)(62 65)(63 66)(64 67)(69 73)(70 74)(71 75)(72 76)(77 81)(78 82)(79 83)(80 84)(85 89)(86 90)(87 91)(88 92)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(1 71)(2 72)(3 69)(4 70)(5 33)(6 34)(7 35)(8 36)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 126)(66 127)(67 128)(68 125)
(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)
(1 115 99 23)(2 116 100 24)(3 113 97 21)(4 114 98 22)(5 81 68 49)(6 82 65 50)(7 83 66 51)(8 84 67 52)(9 117 101 25)(10 118 102 26)(11 119 103 27)(12 120 104 28)(13 121 105 29)(14 122 106 30)(15 123 107 31)(16 124 108 32)(17 125 109 33)(18 126 110 34)(19 127 111 35)(20 128 112 36)(37 85 69 53)(38 86 70 54)(39 87 71 55)(40 88 72 56)(41 89 73 57)(42 90 74 58)(43 91 75 59)(44 92 76 60)(45 93 77 61)(46 94 78 62)(47 95 79 63)(48 96 80 64)

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

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

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

128 conjugacy classes

class 1 2A···2AE4A···4CR
order12···24···4
size11···11···1

128 irreducible representations

dim1111
type+++
imageC1C2C2C4
kernelC23×C42C22×C42C24×C4C23×C4
# reps128396

Matrix representation of C23×C42 in GL5(𝔽5)

40000
04000
00100
00010
00001
,
40000
04000
00100
00010
00004
,
40000
01000
00400
00010
00001
,
30000
03000
00300
00020
00003
,
40000
04000
00400
00020
00001

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,3],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,1] >;

C23×C42 in GAP, Magma, Sage, TeX

C_2^3\times C_4^2
% in TeX

G:=Group("C2^3xC4^2");
// GroupNames label

G:=SmallGroup(128,2150);
// by ID

G=gap.SmallGroup(128,2150);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,456]);
// Polycyclic

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

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