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## G = C32×C8.C22order 288 = 25·32

### Direct product of C32 and C8.C22

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×C8.C22
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×SD16 — C32×C8.C22
 Lower central C1 — C2 — C4 — C32×C8.C22
 Upper central C1 — C3×C6 — C6×C12 — C32×C8.C22

Generators and relations for C32×C8.C22
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 252 in 180 conjugacy classes, 120 normal (24 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, C6 [×4], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×2], Q8, C32, C12 [×8], C12 [×12], C2×C6 [×4], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×C6, C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C12 [×8], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×12], C3×Q8 [×4], C8.C22, C3×C12 [×2], C3×C12 [×3], C62, C62, C3×M4(2) [×4], C3×SD16 [×8], C3×Q16 [×8], C6×Q8 [×4], C3×C4○D4 [×4], C3×C24 [×2], C6×C12, C6×C12 [×2], D4×C32, D4×C32, Q8×C32, Q8×C32 [×2], Q8×C32, C3×C8.C22 [×4], C32×M4(2), C32×SD16 [×2], C32×Q16 [×2], Q8×C3×C6, C32×C4○D4, C32×C8.C22
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C8.C22, C62 [×7], C6×D4 [×4], D4×C32 [×2], C2×C62, C3×C8.C22 [×4], D4×C3×C6, C32×C8.C22

Smallest permutation representation of C32×C8.C22
On 144 points
Generators in S144
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 84 135)(10 85 136)(11 86 129)(12 87 130)(13 88 131)(14 81 132)(15 82 133)(16 83 134)(17 112 142)(18 105 143)(19 106 144)(20 107 137)(21 108 138)(22 109 139)(23 110 140)(24 111 141)(41 102 49)(42 103 50)(43 104 51)(44 97 52)(45 98 53)(46 99 54)(47 100 55)(48 101 56)(57 78 65)(58 79 66)(59 80 67)(60 73 68)(61 74 69)(62 75 70)(63 76 71)(64 77 72)(89 127 119)(90 128 120)(91 121 113)(92 122 114)(93 123 115)(94 124 116)(95 125 117)(96 126 118)
(1 95 42)(2 96 43)(3 89 44)(4 90 45)(5 91 46)(6 92 47)(7 93 48)(8 94 41)(9 143 73)(10 144 74)(11 137 75)(12 138 76)(13 139 77)(14 140 78)(15 141 79)(16 142 80)(17 67 83)(18 68 84)(19 69 85)(20 70 86)(21 71 87)(22 72 88)(23 65 81)(24 66 82)(25 113 54)(26 114 55)(27 115 56)(28 116 49)(29 117 50)(30 118 51)(31 119 52)(32 120 53)(33 122 100)(34 123 101)(35 124 102)(36 125 103)(37 126 104)(38 127 97)(39 128 98)(40 121 99)(57 132 110)(58 133 111)(59 134 112)(60 135 105)(61 136 106)(62 129 107)(63 130 108)(64 131 109)
(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)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 59)(58 62)(61 63)(65 67)(66 70)(69 71)(74 76)(75 79)(78 80)(81 83)(82 86)(85 87)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(106 108)(107 111)(110 112)(114 116)(115 119)(118 120)(122 124)(123 127)(126 128)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)
(1 62)(2 59)(3 64)(4 61)(5 58)(6 63)(7 60)(8 57)(9 123)(10 128)(11 125)(12 122)(13 127)(14 124)(15 121)(16 126)(17 51)(18 56)(19 53)(20 50)(21 55)(22 52)(23 49)(24 54)(25 66)(26 71)(27 68)(28 65)(29 70)(30 67)(31 72)(32 69)(33 76)(34 73)(35 78)(36 75)(37 80)(38 77)(39 74)(40 79)(41 110)(42 107)(43 112)(44 109)(45 106)(46 111)(47 108)(48 105)(81 116)(82 113)(83 118)(84 115)(85 120)(86 117)(87 114)(88 119)(89 131)(90 136)(91 133)(92 130)(93 135)(94 132)(95 129)(96 134)(97 139)(98 144)(99 141)(100 138)(101 143)(102 140)(103 137)(104 142)

G:=sub<Sym(144)| (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,84,135)(10,85,136)(11,86,129)(12,87,130)(13,88,131)(14,81,132)(15,82,133)(16,83,134)(17,112,142)(18,105,143)(19,106,144)(20,107,137)(21,108,138)(22,109,139)(23,110,140)(24,111,141)(41,102,49)(42,103,50)(43,104,51)(44,97,52)(45,98,53)(46,99,54)(47,100,55)(48,101,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(89,127,119)(90,128,120)(91,121,113)(92,122,114)(93,123,115)(94,124,116)(95,125,117)(96,126,118), (1,95,42)(2,96,43)(3,89,44)(4,90,45)(5,91,46)(6,92,47)(7,93,48)(8,94,41)(9,143,73)(10,144,74)(11,137,75)(12,138,76)(13,139,77)(14,140,78)(15,141,79)(16,142,80)(17,67,83)(18,68,84)(19,69,85)(20,70,86)(21,71,87)(22,72,88)(23,65,81)(24,66,82)(25,113,54)(26,114,55)(27,115,56)(28,116,49)(29,117,50)(30,118,51)(31,119,52)(32,120,53)(33,122,100)(34,123,101)(35,124,102)(36,125,103)(37,126,104)(38,127,97)(39,128,98)(40,121,99)(57,132,110)(58,133,111)(59,134,112)(60,135,105)(61,136,106)(62,129,107)(63,130,108)(64,131,109), (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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(81,83)(82,86)(85,87)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(106,108)(107,111)(110,112)(114,116)(115,119)(118,120)(122,124)(123,127)(126,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,123)(10,128)(11,125)(12,122)(13,127)(14,124)(15,121)(16,126)(17,51)(18,56)(19,53)(20,50)(21,55)(22,52)(23,49)(24,54)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(33,76)(34,73)(35,78)(36,75)(37,80)(38,77)(39,74)(40,79)(41,110)(42,107)(43,112)(44,109)(45,106)(46,111)(47,108)(48,105)(81,116)(82,113)(83,118)(84,115)(85,120)(86,117)(87,114)(88,119)(89,131)(90,136)(91,133)(92,130)(93,135)(94,132)(95,129)(96,134)(97,139)(98,144)(99,141)(100,138)(101,143)(102,140)(103,137)(104,142)>;

G:=Group( (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,84,135)(10,85,136)(11,86,129)(12,87,130)(13,88,131)(14,81,132)(15,82,133)(16,83,134)(17,112,142)(18,105,143)(19,106,144)(20,107,137)(21,108,138)(22,109,139)(23,110,140)(24,111,141)(41,102,49)(42,103,50)(43,104,51)(44,97,52)(45,98,53)(46,99,54)(47,100,55)(48,101,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(89,127,119)(90,128,120)(91,121,113)(92,122,114)(93,123,115)(94,124,116)(95,125,117)(96,126,118), (1,95,42)(2,96,43)(3,89,44)(4,90,45)(5,91,46)(6,92,47)(7,93,48)(8,94,41)(9,143,73)(10,144,74)(11,137,75)(12,138,76)(13,139,77)(14,140,78)(15,141,79)(16,142,80)(17,67,83)(18,68,84)(19,69,85)(20,70,86)(21,71,87)(22,72,88)(23,65,81)(24,66,82)(25,113,54)(26,114,55)(27,115,56)(28,116,49)(29,117,50)(30,118,51)(31,119,52)(32,120,53)(33,122,100)(34,123,101)(35,124,102)(36,125,103)(37,126,104)(38,127,97)(39,128,98)(40,121,99)(57,132,110)(58,133,111)(59,134,112)(60,135,105)(61,136,106)(62,129,107)(63,130,108)(64,131,109), (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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(81,83)(82,86)(85,87)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(106,108)(107,111)(110,112)(114,116)(115,119)(118,120)(122,124)(123,127)(126,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,123)(10,128)(11,125)(12,122)(13,127)(14,124)(15,121)(16,126)(17,51)(18,56)(19,53)(20,50)(21,55)(22,52)(23,49)(24,54)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(33,76)(34,73)(35,78)(36,75)(37,80)(38,77)(39,74)(40,79)(41,110)(42,107)(43,112)(44,109)(45,106)(46,111)(47,108)(48,105)(81,116)(82,113)(83,118)(84,115)(85,120)(86,117)(87,114)(88,119)(89,131)(90,136)(91,133)(92,130)(93,135)(94,132)(95,129)(96,134)(97,139)(98,144)(99,141)(100,138)(101,143)(102,140)(103,137)(104,142) );

G=PermutationGroup([(1,36,29),(2,37,30),(3,38,31),(4,39,32),(5,40,25),(6,33,26),(7,34,27),(8,35,28),(9,84,135),(10,85,136),(11,86,129),(12,87,130),(13,88,131),(14,81,132),(15,82,133),(16,83,134),(17,112,142),(18,105,143),(19,106,144),(20,107,137),(21,108,138),(22,109,139),(23,110,140),(24,111,141),(41,102,49),(42,103,50),(43,104,51),(44,97,52),(45,98,53),(46,99,54),(47,100,55),(48,101,56),(57,78,65),(58,79,66),(59,80,67),(60,73,68),(61,74,69),(62,75,70),(63,76,71),(64,77,72),(89,127,119),(90,128,120),(91,121,113),(92,122,114),(93,123,115),(94,124,116),(95,125,117),(96,126,118)], [(1,95,42),(2,96,43),(3,89,44),(4,90,45),(5,91,46),(6,92,47),(7,93,48),(8,94,41),(9,143,73),(10,144,74),(11,137,75),(12,138,76),(13,139,77),(14,140,78),(15,141,79),(16,142,80),(17,67,83),(18,68,84),(19,69,85),(20,70,86),(21,71,87),(22,72,88),(23,65,81),(24,66,82),(25,113,54),(26,114,55),(27,115,56),(28,116,49),(29,117,50),(30,118,51),(31,119,52),(32,120,53),(33,122,100),(34,123,101),(35,124,102),(36,125,103),(37,126,104),(38,127,97),(39,128,98),(40,121,99),(57,132,110),(58,133,111),(59,134,112),(60,135,105),(61,136,106),(62,129,107),(63,130,108),(64,131,109)], [(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),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(57,59),(58,62),(61,63),(65,67),(66,70),(69,71),(74,76),(75,79),(78,80),(81,83),(82,86),(85,87),(89,93),(90,96),(92,94),(97,101),(98,104),(100,102),(106,108),(107,111),(110,112),(114,116),(115,119),(118,120),(122,124),(123,127),(126,128),(129,133),(130,136),(132,134),(137,141),(138,144),(140,142)], [(1,62),(2,59),(3,64),(4,61),(5,58),(6,63),(7,60),(8,57),(9,123),(10,128),(11,125),(12,122),(13,127),(14,124),(15,121),(16,126),(17,51),(18,56),(19,53),(20,50),(21,55),(22,52),(23,49),(24,54),(25,66),(26,71),(27,68),(28,65),(29,70),(30,67),(31,72),(32,69),(33,76),(34,73),(35,78),(36,75),(37,80),(38,77),(39,74),(40,79),(41,110),(42,107),(43,112),(44,109),(45,106),(46,111),(47,108),(48,105),(81,116),(82,113),(83,118),(84,115),(85,120),(86,117),(87,114),(88,119),(89,131),(90,136),(91,133),(92,130),(93,135),(94,132),(95,129),(96,134),(97,139),(98,144),(99,141),(100,138),(101,143),(102,140),(103,137),(104,142)])

99 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 4E 6A ··· 6H 6I ··· 6P 6Q ··· 6X 8A 8B 12A ··· 12P 12Q ··· 12AN 24A ··· 24P order 1 2 2 2 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 4 1 ··· 1 2 2 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C8.C22 C3×C8.C22 kernel C32×C8.C22 C32×M4(2) C32×SD16 C32×Q16 Q8×C3×C6 C32×C4○D4 C3×C8.C22 C3×M4(2) C3×SD16 C3×Q16 C6×Q8 C3×C4○D4 C3×C12 C62 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 8 8 16 16 8 8 1 1 8 8 1 8

Matrix representation of C32×C8.C22 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 64 0 0 0 0 0 0 64 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 1
,
 17 2 0 0 0 0 1 56 0 0 0 0 0 0 5 50 5 28 0 0 68 0 23 45 0 0 22 0 45 50 0 0 51 51 23 23
,
 1 0 0 0 0 0 56 72 0 0 0 0 0 0 1 1 1 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 71 72 72 72 0 0 0 1 0 0

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,1,0,0,0,0,2,56,0,0,0,0,0,0,5,68,22,51,0,0,50,0,0,51,0,0,5,23,45,23,0,0,28,45,50,23],[1,56,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,71,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1,72,0] >;

C32×C8.C22 in GAP, Magma, Sage, TeX

C_3^2\times C_8.C_2^2
% in TeX

G:=Group("C3^2xC8.C2^2");
// GroupNames label

G:=SmallGroup(288,834);
// by ID

G=gap.SmallGroup(288,834);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1016,3110,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations

׿
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