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## G = C32×C4.10D4order 288 = 25·32

### Direct product of C32 and C4.10D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×C12 — C32×M4(2) — C32×C4.10D4
 Lower central C1 — C2 — C22 — C32×C4.10D4
 Upper central C1 — C3×C6 — C6×C12 — C32×C4.10D4

Generators and relations for C32×C4.10D4
G = < a,b,c,d,e | a3=b3=c4=1, d4=c2, e2=dcd-1=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 156 in 114 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], M4(2) [×2], C2×Q8, C3×C6, C3×C6, C24 [×8], C2×C12 [×12], C3×Q8 [×8], C4.10D4, C3×C12 [×2], C3×C12 [×2], C62, C3×M4(2) [×8], C6×Q8 [×4], C3×C24 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], C3×C4.10D4 [×4], C32×M4(2) [×2], Q8×C3×C6, C32×C4.10D4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], C4.10D4, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C6×C12, D4×C32 [×2], C3×C4.10D4 [×4], C32×C22⋊C4, C32×C4.10D4

Smallest permutation representation of C32×C4.10D4
On 144 points
Generators in S144
(1 99 91)(2 100 92)(3 101 93)(4 102 94)(5 103 95)(6 104 96)(7 97 89)(8 98 90)(9 38 138)(10 39 139)(11 40 140)(12 33 141)(13 34 142)(14 35 143)(15 36 144)(16 37 137)(17 125 117)(18 126 118)(19 127 119)(20 128 120)(21 121 113)(22 122 114)(23 123 115)(24 124 116)(25 53 45)(26 54 46)(27 55 47)(28 56 48)(29 49 41)(30 50 42)(31 51 43)(32 52 44)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)(81 111 131)(82 112 132)(83 105 133)(84 106 134)(85 107 135)(86 108 136)(87 109 129)(88 110 130)
(1 19 47)(2 20 48)(3 21 41)(4 22 42)(5 23 43)(6 24 44)(7 17 45)(8 18 46)(9 66 110)(10 67 111)(11 68 112)(12 69 105)(13 70 106)(14 71 107)(15 72 108)(16 65 109)(25 97 125)(26 98 126)(27 99 127)(28 100 128)(29 101 121)(30 102 122)(31 103 123)(32 104 124)(33 61 133)(34 62 134)(35 63 135)(36 64 136)(37 57 129)(38 58 130)(39 59 131)(40 60 132)(49 93 113)(50 94 114)(51 95 115)(52 96 116)(53 89 117)(54 90 118)(55 91 119)(56 92 120)(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 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 83 85 87)(82 88 86 84)(89 91 93 95)(90 96 94 92)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)(113 115 117 119)(114 120 118 116)(121 123 125 127)(122 128 126 124)(129 131 133 135)(130 136 134 132)(137 139 141 143)(138 144 142 140)
(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 60 7 62 5 64 3 58)(2 61 4 59 6 57 8 63)(9 55 11 53 13 51 15 49)(10 52 16 54 14 56 12 50)(17 134 23 136 21 130 19 132)(18 135 20 133 22 131 24 129)(25 142 31 144 29 138 27 140)(26 143 28 141 30 139 32 137)(33 42 39 44 37 46 35 48)(34 43 36 41 38 47 40 45)(65 90 71 92 69 94 67 96)(66 91 68 89 70 95 72 93)(73 102 79 104 77 98 75 100)(74 103 76 101 78 99 80 97)(81 124 87 126 85 128 83 122)(82 125 84 123 86 121 88 127)(105 114 111 116 109 118 107 120)(106 115 108 113 110 119 112 117)

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

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

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

99 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 4C 4D 6A ··· 6H 6I ··· 6P 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AF 24A ··· 24AF order 1 2 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 1 ··· 1 2 2 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 2 2 4 4 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 C4.10D4 C3×C4.10D4 kernel C32×C4.10D4 C32×M4(2) Q8×C3×C6 C3×C4.10D4 C6×C12 C3×M4(2) C6×Q8 C2×C12 C3×C12 C12 C32 C3 # reps 1 2 1 8 4 16 8 32 2 16 1 8

Matrix representation of C32×C4.10D4 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 8 0 0 0 0 0 0 8 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
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 71 0 0 0 0 1 1 0 0 0 0 27 27 0 72 0 0 0 46 1 0
,
 0 27 0 0 0 0 46 0 0 0 0 0 0 0 46 0 71 0 0 0 0 0 1 1 0 0 0 1 27 0 0 0 1 0 46 0
,
 0 46 0 0 0 0 46 0 0 0 0 0 0 0 32 0 12 61 0 0 57 0 0 12 0 0 6 67 57 16 0 0 0 6 16 57

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,27,0,0,0,71,1,27,46,0,0,0,0,0,1,0,0,0,0,72,0],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,46,0,0,1,0,0,0,0,1,0,0,0,71,1,27,46,0,0,0,1,0,0],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,32,57,6,0,0,0,0,0,67,6,0,0,12,0,57,16,0,0,61,12,16,57] >;

C32×C4.10D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4._{10}D_4
% in TeX

G:=Group("C3^2xC4.10D4");
// GroupNames label

G:=SmallGroup(288,319);
// by ID

G=gap.SmallGroup(288,319);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,1016,6304,4548,124]);
// Polycyclic

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

׿
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