Copied to
clipboard

G = C22×C32order 128 = 27

Abelian group of type [2,2,32]

direct product, p-group, abelian, monomial

Aliases: C22×C32, SmallGroup(128,988)

Series: Derived Chief Lower central Upper central Jennings

C1 — C22×C32
C1C2C4C8C16C2×C16C22×C16 — C22×C32
C1 — C22×C32
C1 — C22×C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C22×C32

Generators and relations for C22×C32
 G = < a,b,c | a2=b2=c32=1, ab=ba, ac=ca, bc=cb >

Subgroups: 60, all normal (10 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C8, C8 [×3], C2×C4 [×6], C23, C16, C16 [×3], C2×C8 [×6], C22×C4, C32 [×4], C2×C16 [×6], C22×C8, C2×C32 [×6], C22×C16, C22×C32
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C16 [×4], C2×C8 [×6], C22×C4, C32 [×4], C2×C16 [×6], C22×C8, C2×C32 [×6], C22×C16, C22×C32

Smallest permutation representation of C22×C32
Regular action on 128 points
Generators in S128
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)
(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)| (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106), (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( (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106), (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([(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,127),(23,128),(24,97),(25,98),(26,99),(27,100),(28,101),(29,102),(30,103),(31,104),(32,105),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,81),(32,82),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106)], [(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···2G4A···4H8A···8P16A···16AF32A···32BL
order12···24···48···816···1632···32
size11···11···11···11···11···1

128 irreducible representations

dim1111111111
type+++
imageC1C2C2C4C4C8C8C16C16C32
kernelC22×C32C2×C32C22×C16C2×C16C22×C8C2×C8C22×C4C2×C4C23C22
# reps1616212424864

Matrix representation of C22×C32 in GL3(𝔽97) generated by

9600
0960
0096
,
100
0960
0096
,
2000
0700
0052
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[1,0,0,0,96,0,0,0,96],[20,0,0,0,70,0,0,0,52] >;

C22×C32 in GAP, Magma, Sage, TeX

C_2^2\times C_{32}
% in TeX

G:=Group("C2^2xC32");
// GroupNames label

G:=SmallGroup(128,988);
// by ID

G=gap.SmallGroup(128,988);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,56,80,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^32=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽