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## G = C32×C22⋊C8order 288 = 25·32

### Direct product of C32 and C22⋊C8

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C22⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×C12 — C6×C24 — C32×C22⋊C8
 Lower central C1 — C2 — C32×C22⋊C8
 Upper central C1 — C6×C12 — C32×C22⋊C8

Generators and relations for C32×C22⋊C8
G = < a,b,c,d,e | a3=b3=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 204 in 150 conjugacy classes, 96 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C32, C12 [×8], C12 [×4], C2×C6 [×12], C2×C6 [×8], C2×C8 [×2], C22×C4, C3×C6 [×3], C3×C6 [×2], C24 [×8], C2×C12 [×8], C2×C12 [×8], C22×C6 [×4], C22⋊C8, C3×C12 [×2], C3×C12, C62, C62 [×2], C62 [×2], C2×C24 [×8], C22×C12 [×4], C3×C24 [×2], C6×C12 [×2], C6×C12 [×2], C2×C62, C3×C22⋊C8 [×4], C6×C24 [×2], C2×C6×C12, C32×C22⋊C8
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C2×C8, M4(2), C3×C6 [×3], C24 [×8], C2×C12 [×4], C3×D4 [×8], C22⋊C8, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C2×C24 [×4], C3×M4(2) [×4], C3×C24 [×2], C6×C12, D4×C32 [×2], C3×C22⋊C8 [×4], C32×C22⋊C4, C6×C24, C32×M4(2), C32×C22⋊C8

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

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 4D 4E 4F 6A ··· 6X 6Y ··· 6AN 8A ··· 8H 12A ··· 12AF 12AG ··· 12AV 24A ··· 24BL order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 ··· 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 M4(2) C3×D4 C3×M4(2) kernel C32×C22⋊C8 C6×C24 C2×C6×C12 C3×C22⋊C8 C6×C12 C2×C62 C2×C24 C22×C12 C62 C2×C12 C22×C6 C2×C6 C3×C12 C3×C6 C12 C6 # reps 1 2 1 8 2 2 16 8 8 16 16 64 2 2 16 16

Matrix representation of C32×C22⋊C8 in GL3(𝔽73) generated by

 1 0 0 0 64 0 0 0 64
,
 64 0 0 0 1 0 0 0 1
,
 1 0 0 0 72 0 0 26 1
,
 1 0 0 0 72 0 0 0 72
,
 22 0 0 0 47 71 0 69 26
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[64,0,0,0,1,0,0,0,1],[1,0,0,0,72,26,0,0,1],[1,0,0,0,72,0,0,0,72],[22,0,0,0,47,69,0,71,26] >;

C32×C22⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3^2xC2^2:C8");
// GroupNames label

G:=SmallGroup(288,316);
// by ID

G=gap.SmallGroup(288,316);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

׿
×
𝔽