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G = C24×C8order 128 = 27

Abelian group of type [2,2,2,2,8]

direct product, p-group, abelian, monomial

Aliases: C24×C8, SmallGroup(128,2301)

Series: Derived Chief Lower central Upper central Jennings

C1 — C24×C8
C1C2C4C2×C4C22×C4C23×C4C24×C4 — C24×C8
C1 — C24×C8
C1 — C24×C8
C1C2C2C4 — C24×C8

Generators and relations for C24×C8
 G = < a,b,c,d,e | a2=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 988, all normal (6 characteristic)
C1, C2, C2 [×30], C4, C4 [×15], C22 [×155], C8 [×16], C2×C4 [×120], C23 [×155], C2×C8 [×120], C22×C4 [×140], C24 [×31], C22×C8 [×140], C23×C4 [×30], C25, C23×C8 [×30], C24×C4, C24×C8
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C8 [×16], C2×C4 [×120], C23 [×155], C2×C8 [×120], C22×C4 [×140], C24 [×31], C22×C8 [×140], C23×C4 [×30], C25, C23×C8 [×30], C24×C4, C24×C8

Smallest permutation representation of C24×C8
Regular action on 128 points
Generators in S128
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(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 87)(2 88)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 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 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 73)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 125)(66 126)(67 127)(68 128)(69 121)(70 122)(71 123)(72 124)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)
(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)| (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(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), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,125)(66,126)(67,127)(68,128)(69,121)(70,122)(71,123)(72,124), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120), (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( (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(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), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,125)(66,126)(67,127)(68,128)(69,121)(70,122)(71,123)(72,124), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120), (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([(1,95),(2,96),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(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)], [(1,87),(2,88),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(57,69),(58,70),(59,71),(60,72),(61,65),(62,66),(63,67),(64,68),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(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)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,73),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,125),(66,126),(67,127),(68,128),(69,121),(70,122),(71,123),(72,124)], [(1,103),(2,104),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120)], [(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···2AE4A···4AF8A···8BL
order12···24···48···8
size11···11···11···1

128 irreducible representations

dim111111
type+++
imageC1C2C2C4C4C8
kernelC24×C8C23×C8C24×C4C23×C4C25C24
# reps130130264

Matrix representation of C24×C8 in GL5(𝔽17)

160000
016000
001600
000160
00001
,
10000
01000
001600
000160
000016
,
160000
01000
00100
000160
000016
,
160000
01000
001600
00010
000016
,
80000
02000
001600
00090
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16],[8,0,0,0,0,0,2,0,0,0,0,0,16,0,0,0,0,0,9,0,0,0,0,0,16] >;

C24×C8 in GAP, Magma, Sage, TeX

C_2^4\times C_8
% in TeX

G:=Group("C2^4xC8");
// GroupNames label

G:=SmallGroup(128,2301);
// by ID

G=gap.SmallGroup(128,2301);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=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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