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## G = C23×C42order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23×C42
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C23×C42
 Lower central C1 — C23×C42
 Upper central C1 — C23×C42
 Jennings C1 — C22 — 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 ··· 2AE 4A ··· 4CR order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 1 ··· 1

128 irreducible representations

 dim 1 1 1 1 type + + + image C1 C2 C2 C4 kernel C23×C42 C22×C42 C24×C4 C23×C4 # reps 1 28 3 96

Matrix representation of C23×C42 in GL5(𝔽5)

 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

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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