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## G = C32×C22.D4order 288 = 25·32

### Direct product of C32 and C22.D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C22.D4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=de-1 >

Subgroups: 348 in 234 conjugacy classes, 132 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C22×C6, C22.D4, C3×C12, C62, C62, C62, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×C12, C6×C12, C6×C12, D4×C32, C2×C62, C3×C22.D4, C32×C22⋊C4, C32×C22⋊C4, C32×C4⋊C4, C2×C6×C12, D4×C3×C6, C32×C22.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C4○D4, C3×C6, C3×D4, C22×C6, C22.D4, C62, C6×D4, C3×C4○D4, D4×C32, C2×C62, C3×C22.D4, D4×C3×C6, C32×C4○D4, C32×C22.D4

Smallest permutation representation of C32×C22.D4
On 144 points
Generators in S144
(1 54 46)(2 55 47)(3 56 48)(4 53 45)(5 50 13)(6 51 14)(7 52 15)(8 49 16)(9 42 17)(10 43 18)(11 44 19)(12 41 20)(21 75 29)(22 76 30)(23 73 31)(24 74 32)(25 70 62)(26 71 63)(27 72 64)(28 69 61)(33 66 58)(34 67 59)(35 68 60)(36 65 57)(37 103 95)(38 104 96)(39 101 93)(40 102 94)(77 109 85)(78 110 86)(79 111 87)(80 112 88)(81 128 120)(82 125 117)(83 126 118)(84 127 119)(89 124 116)(90 121 113)(91 122 114)(92 123 115)(97 144 136)(98 141 133)(99 142 134)(100 143 135)(105 140 132)(106 137 129)(107 138 130)(108 139 131)
(1 42 5)(2 43 6)(3 44 7)(4 41 8)(9 13 46)(10 14 47)(11 15 48)(12 16 45)(17 50 54)(18 51 55)(19 52 56)(20 49 53)(21 25 58)(22 26 59)(23 27 60)(24 28 57)(29 62 66)(30 63 67)(31 64 68)(32 61 65)(33 75 70)(34 76 71)(35 73 72)(36 74 69)(37 142 107)(38 143 108)(39 144 105)(40 141 106)(77 81 116)(78 82 113)(79 83 114)(80 84 115)(85 120 124)(86 117 121)(87 118 122)(88 119 123)(89 109 128)(90 110 125)(91 111 126)(92 112 127)(93 97 132)(94 98 129)(95 99 130)(96 100 131)(101 136 140)(102 133 137)(103 134 138)(104 135 139)
(1 80)(2 93)(3 78)(4 95)(5 115)(6 132)(7 113)(8 130)(9 119)(10 136)(11 117)(12 134)(13 123)(14 140)(15 121)(16 138)(17 127)(18 144)(19 125)(20 142)(21 77)(22 94)(23 79)(24 96)(25 81)(26 98)(27 83)(28 100)(29 85)(30 102)(31 87)(32 104)(33 89)(34 106)(35 91)(36 108)(37 53)(38 74)(39 55)(40 76)(41 99)(42 84)(43 97)(44 82)(45 103)(46 88)(47 101)(48 86)(49 107)(50 92)(51 105)(52 90)(54 112)(56 110)(57 131)(58 116)(59 129)(60 114)(61 135)(62 120)(63 133)(64 118)(65 139)(66 124)(67 137)(68 122)(69 143)(70 128)(71 141)(72 126)(73 111)(75 109)
(1 24)(2 21)(3 22)(4 23)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(25 43)(26 44)(27 41)(28 42)(29 47)(30 48)(31 45)(32 46)(33 51)(34 52)(35 49)(36 50)(37 111)(38 112)(39 109)(40 110)(53 73)(54 74)(55 75)(56 76)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 137)(122 138)(123 139)(124 140)(125 141)(126 142)(127 143)(128 144)
(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 23)(4 21)(6 60)(8 58)(10 64)(12 62)(14 68)(16 66)(18 72)(20 70)(25 41)(27 43)(29 45)(31 47)(33 49)(35 51)(37 39)(38 112)(40 110)(53 75)(55 73)(77 79)(78 94)(80 96)(81 83)(82 98)(84 100)(85 87)(86 102)(88 104)(89 91)(90 106)(92 108)(93 95)(97 99)(101 103)(105 107)(109 111)(113 129)(114 116)(115 131)(117 133)(118 120)(119 135)(121 137)(122 124)(123 139)(125 141)(126 128)(127 143)(130 132)(134 136)(138 140)(142 144)

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 kernel C32×C22.D4 C32×C22⋊C4 C32×C4⋊C4 C2×C6×C12 D4×C3×C6 C3×C22.D4 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C62 C3×C6 C2×C6 C6 # reps 1 3 2 1 1 8 24 16 8 8 2 4 16 32

Matrix representation of C32×C22.D4 in GL5(𝔽13)

 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3
,
 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 0 0 0 0 0 1 2 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 5 0 0 0 0 8 8 0 0 0 0 0 12 2 0 0 0 12 1
,
 12 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 1 12

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

C32×C22.D4 in GAP, Magma, Sage, TeX

C_3^2\times C_2^2.D_4
% in TeX

G:=Group("C3^2xC2^2.D4");
// GroupNames label

G:=SmallGroup(288,820);
// by ID

G=gap.SmallGroup(288,820);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,3110,394]);
// Polycyclic

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

׿
×
𝔽