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G = C24×C8  order 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

Derived series
C1 — C24×C8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C24×C4 — C24×C8
Lower central
C1 — C24×C8
Upper central
C1 — C24×C8
Jennings
C1 — C2 — C2 — C4 — 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, C4, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C25, C23×C8, C24×C4, C24×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C25, C23×C8, C24×C4, C24×C8

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

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