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## G = C32×C4⋊1D4order 288 = 25·32

### Direct product of C32 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C2×C62 — D4×C3×C6 — C32×C4⋊1D4
 Lower central C1 — C22 — C32×C4⋊1D4
 Upper central C1 — C62 — C32×C4⋊1D4

Generators and relations for C32×C41D4
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 540 in 324 conjugacy classes, 156 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×6], C22, C22 [×12], C6 [×12], C6 [×16], C2×C4 [×3], D4 [×12], C23 [×4], C32, C12 [×24], C2×C6 [×4], C2×C6 [×48], C42, C2×D4 [×6], C3×C6 [×3], C3×C6 [×4], C2×C12 [×12], C3×D4 [×48], C22×C6 [×16], C41D4, C3×C12 [×6], C62, C62 [×12], C4×C12 [×4], C6×D4 [×24], C6×C12 [×3], D4×C32 [×12], C2×C62 [×4], C3×C41D4 [×4], C122, D4×C3×C6 [×6], C32×C41D4
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×6], C23, C32, C2×C6 [×28], C2×D4 [×3], C3×C6 [×7], C3×D4 [×24], C22×C6 [×4], C41D4, C62 [×7], C6×D4 [×12], D4×C32 [×6], C2×C62, C3×C41D4 [×4], D4×C3×C6 [×3], C32×C41D4

Smallest permutation representation of C32×C41D4
On 144 points
Generators in S144
(1 86 78)(2 87 79)(3 88 80)(4 85 77)(5 122 13)(6 123 14)(7 124 15)(8 121 16)(9 114 17)(10 115 18)(11 116 19)(12 113 20)(21 109 29)(22 110 30)(23 111 31)(24 112 32)(25 142 134)(26 143 135)(27 144 136)(28 141 133)(33 138 130)(34 139 131)(35 140 132)(36 137 129)(37 81 45)(38 82 46)(39 83 47)(40 84 48)(41 73 49)(42 74 50)(43 75 51)(44 76 52)(53 98 90)(54 99 91)(55 100 92)(56 97 89)(57 101 65)(58 102 66)(59 103 67)(60 104 68)(61 93 69)(62 94 70)(63 95 71)(64 96 72)(105 125 117)(106 126 118)(107 127 119)(108 128 120)
(1 74 38)(2 75 39)(3 76 40)(4 73 37)(5 106 114)(6 107 115)(7 108 116)(8 105 113)(9 13 118)(10 14 119)(11 15 120)(12 16 117)(17 122 126)(18 123 127)(19 124 128)(20 121 125)(21 25 130)(22 26 131)(23 27 132)(24 28 129)(29 134 138)(30 135 139)(31 136 140)(32 133 137)(33 109 142)(34 110 143)(35 111 144)(36 112 141)(41 45 77)(42 46 78)(43 47 79)(44 48 80)(49 81 85)(50 82 86)(51 83 87)(52 84 88)(53 70 102)(54 71 103)(55 72 104)(56 69 101)(57 89 93)(58 90 94)(59 91 95)(60 92 96)(61 65 97)(62 66 98)(63 67 99)(64 68 100)
(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 91 22 107)(2 92 23 108)(3 89 24 105)(4 90 21 106)(5 37 58 130)(6 38 59 131)(7 39 60 132)(8 40 57 129)(9 41 62 134)(10 42 63 135)(11 43 64 136)(12 44 61 133)(13 45 66 138)(14 46 67 139)(15 47 68 140)(16 48 65 137)(17 49 70 142)(18 50 71 143)(19 51 72 144)(20 52 69 141)(25 114 73 94)(26 115 74 95)(27 116 75 96)(28 113 76 93)(29 118 77 98)(30 119 78 99)(31 120 79 100)(32 117 80 97)(33 122 81 102)(34 123 82 103)(35 124 83 104)(36 121 84 101)(53 109 126 85)(54 110 127 86)(55 111 128 87)(56 112 125 88)
(1 21)(2 24)(3 23)(4 22)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(25 74)(26 73)(27 76)(28 75)(29 78)(30 77)(31 80)(32 79)(33 82)(34 81)(35 84)(36 83)(37 131)(38 130)(39 129)(40 132)(41 135)(42 134)(43 133)(44 136)(45 139)(46 138)(47 137)(48 140)(49 143)(50 142)(51 141)(52 144)(53 54)(55 56)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(85 110)(86 109)(87 112)(88 111)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 4A ··· 4F 6A ··· 6X 6Y ··· 6BD 12A ··· 12AV order 1 2 2 2 2 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C3 C6 C6 D4 C3×D4 kernel C32×C4⋊1D4 C122 D4×C3×C6 C3×C4⋊1D4 C4×C12 C6×D4 C3×C12 C12 # reps 1 1 6 8 8 48 6 48

Matrix representation of C32×C41D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 6 4 0 0 7 7
,
 4 11 0 0 2 9 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 9 1 0 0 0 0 6 4 0 0 1 7
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,6,7,0,0,4,7],[4,2,0,0,11,9,0,0,0,0,12,0,0,0,0,12],[12,9,0,0,0,1,0,0,0,0,6,1,0,0,4,7] >;

C32×C41D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4\rtimes_1D_4
% in TeX

G:=Group("C3^2xC4:1D4");
// GroupNames label

G:=SmallGroup(288,824);
// by ID

G=gap.SmallGroup(288,824);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,512,3110,772]);
// Polycyclic

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

׿
×
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