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## G = C22×C4×C8order 128 = 27

### Abelian group of type [2,2,4,8]

Aliases: C22×C4×C8, SmallGroup(128,1601)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4×C8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C22×C4×C8
 Lower central C1 — C22×C4×C8
 Upper central C1 — C22×C4×C8
 Jennings C1 — C2 — C2 — C2×C4 — C22×C4×C8

Generators and relations for C22×C4×C8
G = < a,b,c,d | a2=b2=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 380, all normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C24, C4×C8, C2×C42, C22×C8, C23×C4, C23×C4, C2×C4×C8, C22×C42, C23×C8, C22×C4×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C24, C4×C8, C2×C42, C22×C8, C23×C4, C2×C4×C8, C22×C42, C23×C8, C22×C4×C8

Smallest permutation representation of C22×C4×C8
Regular action on 128 points
Generators in S128
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 71)(26 72)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 121)(96 122)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 64)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(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)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 113)(96 114)
(1 95 43 23)(2 96 44 24)(3 89 45 17)(4 90 46 18)(5 91 47 19)(6 92 48 20)(7 93 41 21)(8 94 42 22)(9 66 117 103)(10 67 118 104)(11 68 119 97)(12 69 120 98)(13 70 113 99)(14 71 114 100)(15 72 115 101)(16 65 116 102)(25 34 52 78)(26 35 53 79)(27 36 54 80)(28 37 55 73)(29 38 56 74)(30 39 49 75)(31 40 50 76)(32 33 51 77)(57 123 109 87)(58 124 110 88)(59 125 111 81)(60 126 112 82)(61 127 105 83)(62 128 106 84)(63 121 107 85)(64 122 108 86)
(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)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,71)(26,72)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,121)(96,122), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(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)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (1,95,43,23)(2,96,44,24)(3,89,45,17)(4,90,46,18)(5,91,47,19)(6,92,48,20)(7,93,41,21)(8,94,42,22)(9,66,117,103)(10,67,118,104)(11,68,119,97)(12,69,120,98)(13,70,113,99)(14,71,114,100)(15,72,115,101)(16,65,116,102)(25,34,52,78)(26,35,53,79)(27,36,54,80)(28,37,55,73)(29,38,56,74)(30,39,49,75)(31,40,50,76)(32,33,51,77)(57,123,109,87)(58,124,110,88)(59,125,111,81)(60,126,112,82)(61,127,105,83)(62,128,106,84)(63,121,107,85)(64,122,108,86), (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( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,71)(26,72)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,121)(96,122), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(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)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (1,95,43,23)(2,96,44,24)(3,89,45,17)(4,90,46,18)(5,91,47,19)(6,92,48,20)(7,93,41,21)(8,94,42,22)(9,66,117,103)(10,67,118,104)(11,68,119,97)(12,69,120,98)(13,70,113,99)(14,71,114,100)(15,72,115,101)(16,65,116,102)(25,34,52,78)(26,35,53,79)(27,36,54,80)(28,37,55,73)(29,38,56,74)(30,39,49,75)(31,40,50,76)(32,33,51,77)(57,123,109,87)(58,124,110,88)(59,125,111,81)(60,126,112,82)(61,127,105,83)(62,128,106,84)(63,121,107,85)(64,122,108,86), (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([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,87),(18,88),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,71),(26,72),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(89,123),(90,124),(91,125),(92,126),(93,127),(94,128),(95,121),(96,122)], [(1,70),(2,71),(3,72),(4,65),(5,66),(6,67),(7,68),(8,69),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18),(25,64),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(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),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,113),(96,114)], [(1,95,43,23),(2,96,44,24),(3,89,45,17),(4,90,46,18),(5,91,47,19),(6,92,48,20),(7,93,41,21),(8,94,42,22),(9,66,117,103),(10,67,118,104),(11,68,119,97),(12,69,120,98),(13,70,113,99),(14,71,114,100),(15,72,115,101),(16,65,116,102),(25,34,52,78),(26,35,53,79),(27,36,54,80),(28,37,55,73),(29,38,56,74),(30,39,49,75),(31,40,50,76),(32,33,51,77),(57,123,109,87),(58,124,110,88),(59,125,111,81),(60,126,112,82),(61,127,105,83),(62,128,106,84),(63,121,107,85),(64,122,108,86)], [(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 ··· 2O 4A ··· 4AV 8A ··· 8BL order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 1 ··· 1

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + + image C1 C2 C2 C2 C4 C4 C4 C8 kernel C22×C4×C8 C2×C4×C8 C22×C42 C23×C8 C2×C42 C22×C8 C23×C4 C22×C4 # reps 1 12 1 2 12 32 4 64

Matrix representation of C22×C4×C8 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 13 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 4 0 0 0 0 9 0 0 0 0 15 0 0 0 0 13
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[13,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,9,0,0,0,0,15,0,0,0,0,13] >;

C22×C4×C8 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_8
% in TeX

G:=Group("C2^2xC4xC8");
// GroupNames label

G:=SmallGroup(128,1601);
// by ID

G=gap.SmallGroup(128,1601);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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×
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