G = C32⋊72- 1+4 order 288 = 25·32
2nd semidirect product of C32 and 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial
Aliases: C32⋊72- 1+4, C62.282C23, (C6×Q8)⋊11S3, (C3×Q8).69D6, (C2×C12).172D6, C6.63(S3×C23), (C3×C6).62C24, C12.26D6⋊9C2, C12.59D6⋊11C2, (C6×C12).171C22, C12.114(C22×S3), (C3×C12).133C23, C3⋊4(Q8.15D6), C3⋊Dic3.50C23, C32⋊7D4.4C22, C12⋊S3.34C22, (Q8×C32).33C22, C32⋊4Q8.36C22, (Q8×C3⋊S3)⋊9C2, (Q8×C3×C6)⋊14C2, (C2×Q8)⋊7(C3⋊S3), Q8.15(C2×C3⋊S3), C2.11(C23×C3⋊S3), C4.24(C22×C3⋊S3), (C4×C3⋊S3).48C22, (C2×C3⋊S3).54C23, C22.7(C22×C3⋊S3), (C2×C6).290(C22×S3), (C2×C4).23(C2×C3⋊S3), SmallGroup(288,1012)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊72- 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=1, e2=f2=c2, ab=ba, ac=ca, dad=eae-1=a-1, af=fa, bc=cb, dbd=ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >
Subgroups: 1412 in 438 conjugacy classes, 153 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C32, Dic3, C12, D6, C2×C6, C2×Q8, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, 2- 1+4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, Q8×C32, Q8.15D6, C12.59D6, Q8×C3⋊S3, C12.26D6, Q8×C3×C6, C32⋊72- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, 2- 1+4, C2×C3⋊S3, S3×C23, C22×C3⋊S3, Q8.15D6, C23×C3⋊S3, C32⋊72- 1+4
►On 144 points
Generators in S144
(1 32 6)(2 29 7)(3 30 8)(4 31 5)(9 102 95)(10 103 96)(11 104 93)(12 101 94)(13 100 142)(14 97 143)(15 98 144)(16 99 141)(17 140 58)(18 137 59)(19 138 60)(20 139 57)(21 42 126)(22 43 127)(23 44 128)(24 41 125)(25 71 119)(26 72 120)(27 69 117)(28 70 118)(33 81 113)(34 82 114)(35 83 115)(36 84 116)(37 91 56)(38 92 53)(39 89 54)(40 90 55)(45 52 110)(46 49 111)(47 50 112)(48 51 109)(61 87 80)(62 88 77)(63 85 78)(64 86 79)(65 105 135)(66 106 136)(67 107 133)(68 108 134)(73 132 123)(74 129 124)(75 130 121)(76 131 122)
(1 23 121)(2 24 122)(3 21 123)(4 22 124)(5 127 129)(6 128 130)(7 125 131)(8 126 132)(9 16 55)(10 13 56)(11 14 53)(12 15 54)(17 112 133)(18 109 134)(19 110 135)(20 111 136)(25 62 82)(26 63 83)(27 64 84)(28 61 81)(29 41 76)(30 42 73)(31 43 74)(32 44 75)(33 118 80)(34 119 77)(35 120 78)(36 117 79)(37 103 100)(38 104 97)(39 101 98)(40 102 99)(45 65 138)(46 66 139)(47 67 140)(48 68 137)(49 106 57)(50 107 58)(51 108 59)(52 105 60)(69 86 116)(70 87 113)(71 88 114)(72 85 115)(89 94 144)(90 95 141)(91 96 142)(92 93 143)
(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 4)(2 3)(5 32)(6 31)(7 30)(8 29)(9 12)(10 11)(13 53)(14 56)(15 55)(16 54)(17 136)(18 135)(19 134)(20 133)(21 122)(22 121)(23 124)(24 123)(25 26)(27 28)(33 86)(34 85)(35 88)(36 87)(37 143)(38 142)(39 141)(40 144)(41 132)(42 131)(43 130)(44 129)(45 51)(46 50)(47 49)(48 52)(57 67)(58 66)(59 65)(60 68)(61 84)(62 83)(63 82)(64 81)(69 118)(70 117)(71 120)(72 119)(73 125)(74 128)(75 127)(76 126)(77 115)(78 114)(79 113)(80 116)(89 99)(90 98)(91 97)(92 100)(93 103)(94 102)(95 101)(96 104)(105 137)(106 140)(107 139)(108 138)(109 110)(111 112)
(1 9 3 11)(2 10 4 12)(5 101 7 103)(6 102 8 104)(13 124 15 122)(14 121 16 123)(17 63 19 61)(18 64 20 62)(21 53 23 55)(22 54 24 56)(25 109 27 111)(26 110 28 112)(29 96 31 94)(30 93 32 95)(33 67 35 65)(34 68 36 66)(37 127 39 125)(38 128 40 126)(41 91 43 89)(42 92 44 90)(45 118 47 120)(46 119 48 117)(49 71 51 69)(50 72 52 70)(57 88 59 86)(58 85 60 87)(73 143 75 141)(74 144 76 142)(77 137 79 139)(78 138 80 140)(81 133 83 135)(82 134 84 136)(97 130 99 132)(98 131 100 129)(105 113 107 115)(106 114 108 116)
(1 28 3 26)(2 25 4 27)(5 117 7 119)(6 118 8 120)(9 110 11 112)(10 111 12 109)(13 136 15 134)(14 133 16 135)(17 55 19 53)(18 56 20 54)(21 63 23 61)(22 64 24 62)(29 71 31 69)(30 72 32 70)(33 132 35 130)(34 129 36 131)(37 139 39 137)(38 140 40 138)(41 88 43 86)(42 85 44 87)(45 104 47 102)(46 101 48 103)(49 94 51 96)(50 95 52 93)(57 89 59 91)(58 90 60 92)(65 97 67 99)(66 98 68 100)(73 115 75 113)(74 116 76 114)(77 127 79 125)(78 128 80 126)(81 123 83 121)(82 124 84 122)(105 143 107 141)(106 144 108 142)
G:=sub<Sym(144)| (1,32,6)(2,29,7)(3,30,8)(4,31,5)(9,102,95)(10,103,96)(11,104,93)(12,101,94)(13,100,142)(14,97,143)(15,98,144)(16,99,141)(17,140,58)(18,137,59)(19,138,60)(20,139,57)(21,42,126)(22,43,127)(23,44,128)(24,41,125)(25,71,119)(26,72,120)(27,69,117)(28,70,118)(33,81,113)(34,82,114)(35,83,115)(36,84,116)(37,91,56)(38,92,53)(39,89,54)(40,90,55)(45,52,110)(46,49,111)(47,50,112)(48,51,109)(61,87,80)(62,88,77)(63,85,78)(64,86,79)(65,105,135)(66,106,136)(67,107,133)(68,108,134)(73,132,123)(74,129,124)(75,130,121)(76,131,122), (1,23,121)(2,24,122)(3,21,123)(4,22,124)(5,127,129)(6,128,130)(7,125,131)(8,126,132)(9,16,55)(10,13,56)(11,14,53)(12,15,54)(17,112,133)(18,109,134)(19,110,135)(20,111,136)(25,62,82)(26,63,83)(27,64,84)(28,61,81)(29,41,76)(30,42,73)(31,43,74)(32,44,75)(33,118,80)(34,119,77)(35,120,78)(36,117,79)(37,103,100)(38,104,97)(39,101,98)(40,102,99)(45,65,138)(46,66,139)(47,67,140)(48,68,137)(49,106,57)(50,107,58)(51,108,59)(52,105,60)(69,86,116)(70,87,113)(71,88,114)(72,85,115)(89,94,144)(90,95,141)(91,96,142)(92,93,143), (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), (1,4)(2,3)(5,32)(6,31)(7,30)(8,29)(9,12)(10,11)(13,53)(14,56)(15,55)(16,54)(17,136)(18,135)(19,134)(20,133)(21,122)(22,121)(23,124)(24,123)(25,26)(27,28)(33,86)(34,85)(35,88)(36,87)(37,143)(38,142)(39,141)(40,144)(41,132)(42,131)(43,130)(44,129)(45,51)(46,50)(47,49)(48,52)(57,67)(58,66)(59,65)(60,68)(61,84)(62,83)(63,82)(64,81)(69,118)(70,117)(71,120)(72,119)(73,125)(74,128)(75,127)(76,126)(77,115)(78,114)(79,113)(80,116)(89,99)(90,98)(91,97)(92,100)(93,103)(94,102)(95,101)(96,104)(105,137)(106,140)(107,139)(108,138)(109,110)(111,112), (1,9,3,11)(2,10,4,12)(5,101,7,103)(6,102,8,104)(13,124,15,122)(14,121,16,123)(17,63,19,61)(18,64,20,62)(21,53,23,55)(22,54,24,56)(25,109,27,111)(26,110,28,112)(29,96,31,94)(30,93,32,95)(33,67,35,65)(34,68,36,66)(37,127,39,125)(38,128,40,126)(41,91,43,89)(42,92,44,90)(45,118,47,120)(46,119,48,117)(49,71,51,69)(50,72,52,70)(57,88,59,86)(58,85,60,87)(73,143,75,141)(74,144,76,142)(77,137,79,139)(78,138,80,140)(81,133,83,135)(82,134,84,136)(97,130,99,132)(98,131,100,129)(105,113,107,115)(106,114,108,116), (1,28,3,26)(2,25,4,27)(5,117,7,119)(6,118,8,120)(9,110,11,112)(10,111,12,109)(13,136,15,134)(14,133,16,135)(17,55,19,53)(18,56,20,54)(21,63,23,61)(22,64,24,62)(29,71,31,69)(30,72,32,70)(33,132,35,130)(34,129,36,131)(37,139,39,137)(38,140,40,138)(41,88,43,86)(42,85,44,87)(45,104,47,102)(46,101,48,103)(49,94,51,96)(50,95,52,93)(57,89,59,91)(58,90,60,92)(65,97,67,99)(66,98,68,100)(73,115,75,113)(74,116,76,114)(77,127,79,125)(78,128,80,126)(81,123,83,121)(82,124,84,122)(105,143,107,141)(106,144,108,142)>;
G:=Group( (1,32,6)(2,29,7)(3,30,8)(4,31,5)(9,102,95)(10,103,96)(11,104,93)(12,101,94)(13,100,142)(14,97,143)(15,98,144)(16,99,141)(17,140,58)(18,137,59)(19,138,60)(20,139,57)(21,42,126)(22,43,127)(23,44,128)(24,41,125)(25,71,119)(26,72,120)(27,69,117)(28,70,118)(33,81,113)(34,82,114)(35,83,115)(36,84,116)(37,91,56)(38,92,53)(39,89,54)(40,90,55)(45,52,110)(46,49,111)(47,50,112)(48,51,109)(61,87,80)(62,88,77)(63,85,78)(64,86,79)(65,105,135)(66,106,136)(67,107,133)(68,108,134)(73,132,123)(74,129,124)(75,130,121)(76,131,122), (1,23,121)(2,24,122)(3,21,123)(4,22,124)(5,127,129)(6,128,130)(7,125,131)(8,126,132)(9,16,55)(10,13,56)(11,14,53)(12,15,54)(17,112,133)(18,109,134)(19,110,135)(20,111,136)(25,62,82)(26,63,83)(27,64,84)(28,61,81)(29,41,76)(30,42,73)(31,43,74)(32,44,75)(33,118,80)(34,119,77)(35,120,78)(36,117,79)(37,103,100)(38,104,97)(39,101,98)(40,102,99)(45,65,138)(46,66,139)(47,67,140)(48,68,137)(49,106,57)(50,107,58)(51,108,59)(52,105,60)(69,86,116)(70,87,113)(71,88,114)(72,85,115)(89,94,144)(90,95,141)(91,96,142)(92,93,143), (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), (1,4)(2,3)(5,32)(6,31)(7,30)(8,29)(9,12)(10,11)(13,53)(14,56)(15,55)(16,54)(17,136)(18,135)(19,134)(20,133)(21,122)(22,121)(23,124)(24,123)(25,26)(27,28)(33,86)(34,85)(35,88)(36,87)(37,143)(38,142)(39,141)(40,144)(41,132)(42,131)(43,130)(44,129)(45,51)(46,50)(47,49)(48,52)(57,67)(58,66)(59,65)(60,68)(61,84)(62,83)(63,82)(64,81)(69,118)(70,117)(71,120)(72,119)(73,125)(74,128)(75,127)(76,126)(77,115)(78,114)(79,113)(80,116)(89,99)(90,98)(91,97)(92,100)(93,103)(94,102)(95,101)(96,104)(105,137)(106,140)(107,139)(108,138)(109,110)(111,112), (1,9,3,11)(2,10,4,12)(5,101,7,103)(6,102,8,104)(13,124,15,122)(14,121,16,123)(17,63,19,61)(18,64,20,62)(21,53,23,55)(22,54,24,56)(25,109,27,111)(26,110,28,112)(29,96,31,94)(30,93,32,95)(33,67,35,65)(34,68,36,66)(37,127,39,125)(38,128,40,126)(41,91,43,89)(42,92,44,90)(45,118,47,120)(46,119,48,117)(49,71,51,69)(50,72,52,70)(57,88,59,86)(58,85,60,87)(73,143,75,141)(74,144,76,142)(77,137,79,139)(78,138,80,140)(81,133,83,135)(82,134,84,136)(97,130,99,132)(98,131,100,129)(105,113,107,115)(106,114,108,116), (1,28,3,26)(2,25,4,27)(5,117,7,119)(6,118,8,120)(9,110,11,112)(10,111,12,109)(13,136,15,134)(14,133,16,135)(17,55,19,53)(18,56,20,54)(21,63,23,61)(22,64,24,62)(29,71,31,69)(30,72,32,70)(33,132,35,130)(34,129,36,131)(37,139,39,137)(38,140,40,138)(41,88,43,86)(42,85,44,87)(45,104,47,102)(46,101,48,103)(49,94,51,96)(50,95,52,93)(57,89,59,91)(58,90,60,92)(65,97,67,99)(66,98,68,100)(73,115,75,113)(74,116,76,114)(77,127,79,125)(78,128,80,126)(81,123,83,121)(82,124,84,122)(105,143,107,141)(106,144,108,142) );
G=PermutationGroup([[(1,32,6),(2,29,7),(3,30,8),(4,31,5),(9,102,95),(10,103,96),(11,104,93),(12,101,94),(13,100,142),(14,97,143),(15,98,144),(16,99,141),(17,140,58),(18,137,59),(19,138,60),(20,139,57),(21,42,126),(22,43,127),(23,44,128),(24,41,125),(25,71,119),(26,72,120),(27,69,117),(28,70,118),(33,81,113),(34,82,114),(35,83,115),(36,84,116),(37,91,56),(38,92,53),(39,89,54),(40,90,55),(45,52,110),(46,49,111),(47,50,112),(48,51,109),(61,87,80),(62,88,77),(63,85,78),(64,86,79),(65,105,135),(66,106,136),(67,107,133),(68,108,134),(73,132,123),(74,129,124),(75,130,121),(76,131,122)], [(1,23,121),(2,24,122),(3,21,123),(4,22,124),(5,127,129),(6,128,130),(7,125,131),(8,126,132),(9,16,55),(10,13,56),(11,14,53),(12,15,54),(17,112,133),(18,109,134),(19,110,135),(20,111,136),(25,62,82),(26,63,83),(27,64,84),(28,61,81),(29,41,76),(30,42,73),(31,43,74),(32,44,75),(33,118,80),(34,119,77),(35,120,78),(36,117,79),(37,103,100),(38,104,97),(39,101,98),(40,102,99),(45,65,138),(46,66,139),(47,67,140),(48,68,137),(49,106,57),(50,107,58),(51,108,59),(52,105,60),(69,86,116),(70,87,113),(71,88,114),(72,85,115),(89,94,144),(90,95,141),(91,96,142),(92,93,143)], [(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)], [(1,4),(2,3),(5,32),(6,31),(7,30),(8,29),(9,12),(10,11),(13,53),(14,56),(15,55),(16,54),(17,136),(18,135),(19,134),(20,133),(21,122),(22,121),(23,124),(24,123),(25,26),(27,28),(33,86),(34,85),(35,88),(36,87),(37,143),(38,142),(39,141),(40,144),(41,132),(42,131),(43,130),(44,129),(45,51),(46,50),(47,49),(48,52),(57,67),(58,66),(59,65),(60,68),(61,84),(62,83),(63,82),(64,81),(69,118),(70,117),(71,120),(72,119),(73,125),(74,128),(75,127),(76,126),(77,115),(78,114),(79,113),(80,116),(89,99),(90,98),(91,97),(92,100),(93,103),(94,102),(95,101),(96,104),(105,137),(106,140),(107,139),(108,138),(109,110),(111,112)], [(1,9,3,11),(2,10,4,12),(5,101,7,103),(6,102,8,104),(13,124,15,122),(14,121,16,123),(17,63,19,61),(18,64,20,62),(21,53,23,55),(22,54,24,56),(25,109,27,111),(26,110,28,112),(29,96,31,94),(30,93,32,95),(33,67,35,65),(34,68,36,66),(37,127,39,125),(38,128,40,126),(41,91,43,89),(42,92,44,90),(45,118,47,120),(46,119,48,117),(49,71,51,69),(50,72,52,70),(57,88,59,86),(58,85,60,87),(73,143,75,141),(74,144,76,142),(77,137,79,139),(78,138,80,140),(81,133,83,135),(82,134,84,136),(97,130,99,132),(98,131,100,129),(105,113,107,115),(106,114,108,116)], [(1,28,3,26),(2,25,4,27),(5,117,7,119),(6,118,8,120),(9,110,11,112),(10,111,12,109),(13,136,15,134),(14,133,16,135),(17,55,19,53),(18,56,20,54),(21,63,23,61),(22,64,24,62),(29,71,31,69),(30,72,32,70),(33,132,35,130),(34,129,36,131),(37,139,39,137),(38,140,40,138),(41,88,43,86),(42,85,44,87),(45,104,47,102),(46,101,48,103),(49,94,51,96),(50,95,52,93),(57,89,59,91),(58,90,60,92),(65,97,67,99),(66,98,68,100),(73,115,75,113),(74,116,76,114),(77,127,79,125),(78,128,80,126),(81,123,83,121),(82,124,84,122),(105,143,107,141),(106,144,108,142)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2- 1+4 | Q8.15D6 |
| kernel | C32⋊72- 1+4 | C12.59D6 | Q8×C3⋊S3 | C12.26D6 | Q8×C3×C6 | C6×Q8 | C2×C12 | C3×Q8 | C32 | C3 |
| # reps | 1 | 6 | 4 | 4 | 1 | 4 | 12 | 16 | 1 | 8 |
Matrix representation of C32⋊72- 1+4 ►in GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 | 3 | 0 |
| 0 | 0 | 1 | 10 | 0 | 3 |
| 0 | 0 | 3 | 7 | 2 | 1 |
| 0 | 0 | 6 | 10 | 12 | 3 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 10 | 8 |
| 0 | 0 | 10 | 1 | 5 | 3 |
| 0 | 0 | 7 | 3 | 1 | 10 |
| 0 | 0 | 10 | 6 | 11 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 9 | 2 |
| 0 | 0 | 2 | 0 | 11 | 4 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 7 | 12 |
| 0 | 0 | 10 | 11 | 1 | 6 |
| 0 | 0 | 5 | 0 | 5 | 10 |
| 0 | 0 | 0 | 5 | 3 | 2 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,1,3,6,0,0,12,10,7,10,0,0,3,0,2,12,0,0,0,3,1,3],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,10,7,10,0,0,11,1,3,6,0,0,10,5,1,11,0,0,8,3,10,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,9,11,0,2,0,0,2,4,2,0,0,0,0,0,9,11,0,0,0,0,2,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,10,5,0,0,0,3,11,0,5,0,0,7,1,5,3,0,0,12,6,10,2] >;
C32⋊72- 1+4 in GAP, Magma, Sage, TeX
C_3^2\rtimes_72_-^{1+4} % in TeX
G:=Group("C3^2:7ES-(2,2)"); // GroupNames label
G:=SmallGroup(288,1012);
// by ID
G=gap.SmallGroup(288,1012);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,2693,9414]);
// 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,d*a*d=e*a*e^-1=a^-1,a*f=f*a,b*c=c*b,d*b*d=e*b*e^-1=b^-1,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