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## G = C32×C4○D8order 288 = 25·32

### Direct product of C32 and C4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×C4○D8
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×D8 — C32×C4○D8
 Lower central C1 — C2 — C4 — C32×C4○D8
 Upper central C1 — C3×C12 — C6×C12 — C32×C4○D8

Generators and relations for C32×C4○D8
G = < a,b,c,d,e | a3=b3=c4=e2=1, d4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >

Subgroups: 276 in 186 conjugacy classes, 120 normal (24 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], C6 [×4], C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], C2×C6 [×8], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×C6, C3×C6 [×3], C24 [×8], C2×C12 [×4], C2×C12 [×8], C3×D4 [×8], C3×D4 [×8], C3×Q8 [×8], C4○D8, C3×C12 [×2], C3×C12 [×2], C62, C62 [×2], C2×C24 [×4], C3×D8 [×4], C3×SD16 [×8], C3×Q16 [×4], C3×C4○D4 [×8], C3×C24 [×2], C6×C12, C6×C12 [×2], D4×C32 [×2], D4×C32 [×2], Q8×C32 [×2], C3×C4○D8 [×4], C6×C24, C32×D8, C32×SD16 [×2], C32×Q16, C32×C4○D4 [×2], C32×C4○D8
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], C4○D8, C62 [×7], C6×D4 [×4], D4×C32 [×2], C2×C62, C3×C4○D8 [×4], D4×C3×C6, C32×C4○D8

Smallest permutation representation of C32×C4○D8
On 144 points
Generators in S144
(1 38 30)(2 39 31)(3 40 32)(4 33 25)(5 34 26)(6 35 27)(7 36 28)(8 37 29)(9 131 86)(10 132 87)(11 133 88)(12 134 81)(13 135 82)(14 136 83)(15 129 84)(16 130 85)(17 112 144)(18 105 137)(19 106 138)(20 107 139)(21 108 140)(22 109 141)(23 110 142)(24 111 143)(41 98 49)(42 99 50)(43 100 51)(44 101 52)(45 102 53)(46 103 54)(47 104 55)(48 97 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)(89 121 113)(90 122 114)(91 123 115)(92 124 116)(93 125 117)(94 126 118)(95 127 119)(96 128 120)
(1 95 46)(2 96 47)(3 89 48)(4 90 41)(5 91 42)(6 92 43)(7 93 44)(8 94 45)(9 20 66)(10 21 67)(11 22 68)(12 23 69)(13 24 70)(14 17 71)(15 18 72)(16 19 65)(25 114 49)(26 115 50)(27 116 51)(28 117 52)(29 118 53)(30 119 54)(31 120 55)(32 113 56)(33 122 98)(34 123 99)(35 124 100)(36 125 101)(37 126 102)(38 127 103)(39 128 104)(40 121 97)(57 130 106)(58 131 107)(59 132 108)(60 133 109)(61 134 110)(62 135 111)(63 136 112)(64 129 105)(73 88 141)(74 81 142)(75 82 143)(76 83 144)(77 84 137)(78 85 138)(79 86 139)(80 87 140)
(1 58 5 62)(2 59 6 63)(3 60 7 64)(4 61 8 57)(9 115 13 119)(10 116 14 120)(11 117 15 113)(12 118 16 114)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(25 69 29 65)(26 70 30 66)(27 71 31 67)(28 72 32 68)(33 74 37 78)(34 75 38 79)(35 76 39 80)(36 77 40 73)(41 110 45 106)(42 111 46 107)(43 112 47 108)(44 105 48 109)(81 126 85 122)(82 127 86 123)(83 128 87 124)(84 121 88 125)(89 133 93 129)(90 134 94 130)(91 135 95 131)(92 136 96 132)(97 141 101 137)(98 142 102 138)(99 143 103 139)(100 144 104 140)
(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)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 24)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 66)(67 72)(68 71)(69 70)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C4○D8 C3×C4○D8 kernel C32×C4○D8 C6×C24 C32×D8 C32×SD16 C32×Q16 C32×C4○D4 C3×C4○D8 C2×C24 C3×D8 C3×SD16 C3×Q16 C3×C4○D4 C3×C12 C62 C12 C2×C6 C32 C3 # reps 1 1 1 2 1 2 8 8 8 16 8 16 1 1 8 8 4 32

Matrix representation of C32×C4○D8 in GL3(𝔽73) generated by

 64 0 0 0 8 0 0 0 8
,
 8 0 0 0 8 0 0 0 8
,
 72 0 0 0 46 0 0 0 46
,
 72 0 0 0 57 16 0 57 57
,
 1 0 0 0 57 16 0 16 16
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[8,0,0,0,8,0,0,0,8],[72,0,0,0,46,0,0,0,46],[72,0,0,0,57,57,0,16,57],[1,0,0,0,57,16,0,16,16] >;

C32×C4○D8 in GAP, Magma, Sage, TeX

C_3^2\times C_4\circ D_8
% in TeX

G:=Group("C3^2xC4oD8");
// GroupNames label

G:=SmallGroup(288,832);
// by ID

G=gap.SmallGroup(288,832);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,772,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^2=1,d^4=c^2,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,e*d*e=c^2*d^3>;
// generators/relations

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