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## G = C32×C8○D4order 288 = 25·32

### Direct product of C32 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C8○D4
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C3×C24 — C6×C24 — C32×C8○D4
 Lower central C1 — C2 — C32×C8○D4
 Upper central C1 — C3×C24 — C32×C8○D4

Generators and relations for C32×C8○D4
G = < a,b,c,d,e | a3=b3=c8=e2=1, d2=c4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

Subgroups: 204 in 186 conjugacy classes, 168 normal (14 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×3], C6 [×4], C6 [×12], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C32, C12 [×16], C2×C6 [×12], C2×C8 [×3], M4(2) [×3], C4○D4, C3×C6, C3×C6 [×3], C24 [×16], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C8○D4, C3×C12, C3×C12 [×3], C62 [×3], C2×C24 [×12], C3×M4(2) [×12], C3×C4○D4 [×4], C3×C24, C3×C24 [×3], C6×C12 [×3], D4×C32 [×3], Q8×C32, C3×C8○D4 [×4], C6×C24 [×3], C32×M4(2) [×3], C32×C4○D4, C32×C8○D4
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], C8○D4, C3×C12 [×4], C62 [×7], C22×C12 [×4], C6×C12 [×6], C2×C62, C3×C8○D4 [×4], C2×C6×C12, C32×C8○D4

Smallest permutation representation of C32×C8○D4
On 144 points
Generators in S144
(1 97 89)(2 98 90)(3 99 91)(4 100 92)(5 101 93)(6 102 94)(7 103 95)(8 104 96)(9 34 144)(10 35 137)(11 36 138)(12 37 139)(13 38 140)(14 39 141)(15 40 142)(16 33 143)(17 127 119)(18 128 120)(19 121 113)(20 122 114)(21 123 115)(22 124 116)(23 125 117)(24 126 118)(25 55 47)(26 56 48)(27 49 41)(28 50 42)(29 51 43)(30 52 44)(31 53 45)(32 54 46)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)(81 109 131)(82 110 132)(83 111 133)(84 112 134)(85 105 135)(86 106 136)(87 107 129)(88 108 130)
(1 17 47)(2 18 48)(3 19 41)(4 20 42)(5 21 43)(6 22 44)(7 23 45)(8 24 46)(9 72 106)(10 65 107)(11 66 108)(12 67 109)(13 68 110)(14 69 111)(15 70 112)(16 71 105)(25 97 127)(26 98 128)(27 99 121)(28 100 122)(29 101 123)(30 102 124)(31 103 125)(32 104 126)(33 63 135)(34 64 136)(35 57 129)(36 58 130)(37 59 131)(38 60 132)(39 61 133)(40 62 134)(49 91 113)(50 92 114)(51 93 115)(52 94 116)(53 95 117)(54 96 118)(55 89 119)(56 90 120)(73 83 141)(74 84 142)(75 85 143)(76 86 144)(77 87 137)(78 88 138)(79 81 139)(80 82 140)
(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 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 103 101 99)(98 104 102 100)(105 107 109 111)(106 108 110 112)(113 119 117 115)(114 120 118 116)(121 127 125 123)(122 128 126 124)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(17 133)(18 134)(19 135)(20 136)(21 129)(22 130)(23 131)(24 132)(25 141)(26 142)(27 143)(28 144)(29 137)(30 138)(31 139)(32 140)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(65 93)(66 94)(67 95)(68 96)(69 89)(70 90)(71 91)(72 92)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)

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

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

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

180 conjugacy classes

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

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 C8○D4 C3×C8○D4 kernel C32×C8○D4 C6×C24 C32×M4(2) C32×C4○D4 C3×C8○D4 D4×C32 Q8×C32 C2×C24 C3×M4(2) C3×C4○D4 C3×D4 C3×Q8 C32 C3 # reps 1 3 3 1 8 6 2 24 24 8 48 16 4 32

Matrix representation of C32×C8○D4 in GL3(𝔽73) generated by

 64 0 0 0 1 0 0 0 1
,
 64 0 0 0 8 0 0 0 8
,
 46 0 0 0 22 0 0 0 22
,
 1 0 0 0 27 0 0 61 46
,
 1 0 0 0 12 54 0 69 61
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[64,0,0,0,8,0,0,0,8],[46,0,0,0,22,0,0,0,22],[1,0,0,0,27,61,0,0,46],[1,0,0,0,12,69,0,54,61] >;

C32×C8○D4 in GAP, Magma, Sage, TeX

C_3^2\times C_8\circ D_4
% in TeX

G:=Group("C3^2xC8oD4");
// GroupNames label

G:=SmallGroup(288,828);
// by ID

G=gap.SmallGroup(288,828);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1563,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=e^2=1,d^2=c^4,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^4*d>;
// generators/relations

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