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## G = C32×2- 1+4order 288 = 25·32

### Direct product of C32 and 2- 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×2- 1+4
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32 — C32×C4○D4 — C32×2- 1+4
 Lower central C1 — C2 — C32×2- 1+4
 Upper central C1 — C3×C6 — C32×2- 1+4

Generators and relations for C32×2- 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=1, e2=f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 468 in 438 conjugacy classes, 408 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C2×Q8, C4○D4, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, 2- 1+4, C3×C12, C62, C6×Q8, C3×C4○D4, C6×C12, D4×C32, Q8×C32, C3×2- 1+4, Q8×C3×C6, C32×C4○D4, C32×2- 1+4
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, 2- 1+4, C62, C23×C6, C2×C62, C3×2- 1+4, C22×C62, C32×2- 1+4

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

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

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

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

153 conjugacy classes

 class 1 2A 2B ··· 2F 3A ··· 3H 4A ··· 4J 6A ··· 6H 6I ··· 6AV 12A ··· 12CB order 1 2 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

153 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + - image C1 C2 C2 C3 C6 C6 2- 1+4 C3×2- 1+4 kernel C32×2- 1+4 Q8×C3×C6 C32×C4○D4 C3×2- 1+4 C6×Q8 C3×C4○D4 C32 C3 # reps 1 5 10 8 40 80 1 8

Matrix representation of C32×2- 1+4 in GL5(𝔽13)

 3 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 0 10 5 0 0 0 11 3 0 10 5 0 0 0 11 3 0 0
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 5 0 0 0 0 6 8 0 0 0 0 0 8 0 0 0 0 7 5
,
 1 0 0 0 0 0 2 1 0 0 0 8 11 0 0 0 0 0 11 12 0 0 0 5 2

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,10,11,0,0,0,5,3,0,10,11,0,0,0,5,3,0,0],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,5,6,0,0,0,0,8,0,0,0,0,0,8,7,0,0,0,0,5],[1,0,0,0,0,0,2,8,0,0,0,1,11,0,0,0,0,0,11,5,0,0,0,12,2] >;

C32×2- 1+4 in GAP, Magma, Sage, TeX

C_3^2\times 2_-^{1+4}
% in TeX

G:=Group("C3^2xES-(2,2)");
// GroupNames label

G:=SmallGroup(288,1023);
// by ID

G=gap.SmallGroup(288,1023);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045,1016,1563,772,4259]);
// Polycyclic

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

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