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## G = C25×C4order 128 = 27

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

Aliases: C25×C4, SmallGroup(128,2319)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C25×C4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C26 — C25×C4
 Lower central C1 — C25×C4
 Upper central C1 — C25×C4
 Jennings C1 — C2 — C25×C4

Generators and relations for C25×C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 5276, all normal (4 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, C25, C24×C4, C26, C25×C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, C25, C24×C4, C26, C25×C4

Smallest permutation representation of C25×C4
Regular action on 128 points
Generators in S128
(1 87)(2 88)(3 85)(4 86)(5 22)(6 23)(7 24)(8 21)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 67)(50 68)(51 65)(52 66)(69 81)(70 82)(71 83)(72 84)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)
(1 79)(2 80)(3 77)(4 78)(5 30)(6 31)(7 32)(8 29)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(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 67)(58 68)(59 65)(60 66)(69 89)(70 90)(71 91)(72 92)(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 75)(2 76)(3 73)(4 74)(5 34)(6 35)(7 36)(8 33)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(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 67)(62 68)(63 65)(64 66)(69 93)(70 94)(71 95)(72 96)(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 11)(2 12)(3 9)(4 10)(5 70)(6 71)(7 72)(8 69)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 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)(65 127)(66 128)(67 125)(68 126)
(1 99)(2 100)(3 97)(4 98)(5 68)(6 65)(7 66)(8 67)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(69 125)(70 126)(71 127)(72 128)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)
(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)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2BK 4A ··· 4BL 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 C25×C4 C24×C4 C26 C25 # reps 1 62 1 64

Matrix representation of C25×C4 in GL6(𝔽5)

 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3] >;

C25×C4 in GAP, Magma, Sage, TeX

C_2^5\times C_4
% in TeX

G:=Group("C2^5xC4");
// GroupNames label

G:=SmallGroup(128,2319);
// by ID

G=gap.SmallGroup(128,2319);
# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,448]);
// Polycyclic

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

׿
×
𝔽