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 [×5], C3 [×4], C4 [×6], C4 [×4], C22, C22 [×4], S3 [×16], C6 [×4], C6 [×4], C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C32, Dic3 [×16], C12 [×24], D6 [×16], C2×C6 [×4], C2×Q8, C2×Q8 [×4], C4○D4 [×10], C3⋊S3 [×4], C3×C6, C3×C6, Dic6 [×24], C4×S3 [×48], D12 [×24], C3⋊D4 [×16], C2×C12 [×12], C3×Q8 [×16], 2- 1+4, C3⋊Dic3 [×4], C3×C12 [×6], C2×C3⋊S3 [×4], C62, C4○D12 [×24], S3×Q8 [×16], Q8⋊3S3 [×16], C6×Q8 [×4], C32⋊4Q8 [×6], C4×C3⋊S3 [×12], C12⋊S3 [×6], C32⋊7D4 [×4], C6×C12 [×3], Q8×C32 [×4], Q8.15D6 [×4], C12.59D6 [×6], Q8×C3⋊S3 [×4], C12.26D6 [×4], Q8×C3×C6, C32⋊72- 1+4
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C24, C3⋊S3, C22×S3 [×28], 2- 1+4, C2×C3⋊S3 [×7], S3×C23 [×4], C22×C3⋊S3 [×7], Q8.15D6 [×4], C23×C3⋊S3, C32⋊72- 1+4
►On 144 points
Generators in S144
(1 32 76)(2 29 73)(3 30 74)(4 31 75)(5 90 102)(6 91 103)(7 92 104)(8 89 101)(9 55 126)(10 56 127)(11 53 128)(12 54 125)(13 100 142)(14 97 143)(15 98 144)(16 99 141)(17 140 112)(18 137 109)(19 138 110)(20 139 111)(21 44 121)(22 41 122)(23 42 123)(24 43 124)(25 114 86)(26 115 87)(27 116 88)(28 113 85)(33 81 63)(34 82 64)(35 83 61)(36 84 62)(37 95 129)(38 96 130)(39 93 131)(40 94 132)(45 65 105)(46 66 106)(47 67 107)(48 68 108)(49 57 136)(50 58 133)(51 59 134)(52 60 135)(69 117 77)(70 118 78)(71 119 79)(72 120 80)
(1 96 121)(2 93 122)(3 94 123)(4 95 124)(5 141 9)(6 142 10)(7 143 11)(8 144 12)(13 56 91)(14 53 92)(15 54 89)(16 55 90)(17 58 47)(18 59 48)(19 60 45)(20 57 46)(21 32 130)(22 29 131)(23 30 132)(24 31 129)(25 36 117)(26 33 118)(27 34 119)(28 35 120)(37 43 75)(38 44 76)(39 41 73)(40 42 74)(49 106 111)(50 107 112)(51 108 109)(52 105 110)(61 72 85)(62 69 86)(63 70 87)(64 71 88)(65 138 135)(66 139 136)(67 140 133)(68 137 134)(77 114 84)(78 115 81)(79 116 82)(80 113 83)(97 128 104)(98 125 101)(99 126 102)(100 127 103)
(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)(5 101)(6 104)(7 103)(8 102)(9 98)(10 97)(11 100)(12 99)(13 53)(14 56)(15 55)(16 54)(17 46)(18 45)(19 48)(20 47)(21 37)(22 40)(23 39)(24 38)(25 63)(26 62)(27 61)(28 64)(29 74)(30 73)(31 76)(32 75)(33 86)(34 85)(35 88)(36 87)(41 132)(42 131)(43 130)(44 129)(49 133)(50 136)(51 135)(52 134)(57 58)(59 60)(65 109)(66 112)(67 111)(68 110)(69 118)(70 117)(71 120)(72 119)(77 78)(79 80)(81 114)(82 113)(83 116)(84 115)(89 90)(91 92)(93 123)(94 122)(95 121)(96 124)(105 137)(106 140)(107 139)(108 138)(125 141)(126 144)(127 143)(128 142)
(1 90 3 92)(2 91 4 89)(5 30 7 32)(6 31 8 29)(9 132 11 130)(10 129 12 131)(13 124 15 122)(14 121 16 123)(17 113 19 115)(18 114 20 116)(21 141 23 143)(22 142 24 144)(25 139 27 137)(26 140 28 138)(33 67 35 65)(34 68 36 66)(37 125 39 127)(38 126 40 128)(41 100 43 98)(42 97 44 99)(45 81 47 83)(46 82 48 84)(49 71 51 69)(50 72 52 70)(53 96 55 94)(54 93 56 95)(57 79 59 77)(58 80 60 78)(61 105 63 107)(62 106 64 108)(73 103 75 101)(74 104 76 102)(85 110 87 112)(86 111 88 109)(117 136 119 134)(118 133 120 135)
(1 78 3 80)(2 79 4 77)(5 52 7 50)(6 49 8 51)(9 110 11 112)(10 111 12 109)(13 46 15 48)(14 47 16 45)(17 55 19 53)(18 56 20 54)(21 63 23 61)(22 64 24 62)(25 39 27 37)(26 40 28 38)(29 71 31 69)(30 72 32 70)(33 42 35 44)(34 43 36 41)(57 89 59 91)(58 90 60 92)(65 97 67 99)(66 98 68 100)(73 119 75 117)(74 120 76 118)(81 123 83 121)(82 124 84 122)(85 130 87 132)(86 131 88 129)(93 116 95 114)(94 113 96 115)(101 134 103 136)(102 135 104 133)(105 143 107 141)(106 144 108 142)(125 137 127 139)(126 138 128 140)
G:=sub<Sym(144)| (1,32,76)(2,29,73)(3,30,74)(4,31,75)(5,90,102)(6,91,103)(7,92,104)(8,89,101)(9,55,126)(10,56,127)(11,53,128)(12,54,125)(13,100,142)(14,97,143)(15,98,144)(16,99,141)(17,140,112)(18,137,109)(19,138,110)(20,139,111)(21,44,121)(22,41,122)(23,42,123)(24,43,124)(25,114,86)(26,115,87)(27,116,88)(28,113,85)(33,81,63)(34,82,64)(35,83,61)(36,84,62)(37,95,129)(38,96,130)(39,93,131)(40,94,132)(45,65,105)(46,66,106)(47,67,107)(48,68,108)(49,57,136)(50,58,133)(51,59,134)(52,60,135)(69,117,77)(70,118,78)(71,119,79)(72,120,80), (1,96,121)(2,93,122)(3,94,123)(4,95,124)(5,141,9)(6,142,10)(7,143,11)(8,144,12)(13,56,91)(14,53,92)(15,54,89)(16,55,90)(17,58,47)(18,59,48)(19,60,45)(20,57,46)(21,32,130)(22,29,131)(23,30,132)(24,31,129)(25,36,117)(26,33,118)(27,34,119)(28,35,120)(37,43,75)(38,44,76)(39,41,73)(40,42,74)(49,106,111)(50,107,112)(51,108,109)(52,105,110)(61,72,85)(62,69,86)(63,70,87)(64,71,88)(65,138,135)(66,139,136)(67,140,133)(68,137,134)(77,114,84)(78,115,81)(79,116,82)(80,113,83)(97,128,104)(98,125,101)(99,126,102)(100,127,103), (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)(5,101)(6,104)(7,103)(8,102)(9,98)(10,97)(11,100)(12,99)(13,53)(14,56)(15,55)(16,54)(17,46)(18,45)(19,48)(20,47)(21,37)(22,40)(23,39)(24,38)(25,63)(26,62)(27,61)(28,64)(29,74)(30,73)(31,76)(32,75)(33,86)(34,85)(35,88)(36,87)(41,132)(42,131)(43,130)(44,129)(49,133)(50,136)(51,135)(52,134)(57,58)(59,60)(65,109)(66,112)(67,111)(68,110)(69,118)(70,117)(71,120)(72,119)(77,78)(79,80)(81,114)(82,113)(83,116)(84,115)(89,90)(91,92)(93,123)(94,122)(95,121)(96,124)(105,137)(106,140)(107,139)(108,138)(125,141)(126,144)(127,143)(128,142), (1,90,3,92)(2,91,4,89)(5,30,7,32)(6,31,8,29)(9,132,11,130)(10,129,12,131)(13,124,15,122)(14,121,16,123)(17,113,19,115)(18,114,20,116)(21,141,23,143)(22,142,24,144)(25,139,27,137)(26,140,28,138)(33,67,35,65)(34,68,36,66)(37,125,39,127)(38,126,40,128)(41,100,43,98)(42,97,44,99)(45,81,47,83)(46,82,48,84)(49,71,51,69)(50,72,52,70)(53,96,55,94)(54,93,56,95)(57,79,59,77)(58,80,60,78)(61,105,63,107)(62,106,64,108)(73,103,75,101)(74,104,76,102)(85,110,87,112)(86,111,88,109)(117,136,119,134)(118,133,120,135), (1,78,3,80)(2,79,4,77)(5,52,7,50)(6,49,8,51)(9,110,11,112)(10,111,12,109)(13,46,15,48)(14,47,16,45)(17,55,19,53)(18,56,20,54)(21,63,23,61)(22,64,24,62)(25,39,27,37)(26,40,28,38)(29,71,31,69)(30,72,32,70)(33,42,35,44)(34,43,36,41)(57,89,59,91)(58,90,60,92)(65,97,67,99)(66,98,68,100)(73,119,75,117)(74,120,76,118)(81,123,83,121)(82,124,84,122)(85,130,87,132)(86,131,88,129)(93,116,95,114)(94,113,96,115)(101,134,103,136)(102,135,104,133)(105,143,107,141)(106,144,108,142)(125,137,127,139)(126,138,128,140)>;
G:=Group( (1,32,76)(2,29,73)(3,30,74)(4,31,75)(5,90,102)(6,91,103)(7,92,104)(8,89,101)(9,55,126)(10,56,127)(11,53,128)(12,54,125)(13,100,142)(14,97,143)(15,98,144)(16,99,141)(17,140,112)(18,137,109)(19,138,110)(20,139,111)(21,44,121)(22,41,122)(23,42,123)(24,43,124)(25,114,86)(26,115,87)(27,116,88)(28,113,85)(33,81,63)(34,82,64)(35,83,61)(36,84,62)(37,95,129)(38,96,130)(39,93,131)(40,94,132)(45,65,105)(46,66,106)(47,67,107)(48,68,108)(49,57,136)(50,58,133)(51,59,134)(52,60,135)(69,117,77)(70,118,78)(71,119,79)(72,120,80), (1,96,121)(2,93,122)(3,94,123)(4,95,124)(5,141,9)(6,142,10)(7,143,11)(8,144,12)(13,56,91)(14,53,92)(15,54,89)(16,55,90)(17,58,47)(18,59,48)(19,60,45)(20,57,46)(21,32,130)(22,29,131)(23,30,132)(24,31,129)(25,36,117)(26,33,118)(27,34,119)(28,35,120)(37,43,75)(38,44,76)(39,41,73)(40,42,74)(49,106,111)(50,107,112)(51,108,109)(52,105,110)(61,72,85)(62,69,86)(63,70,87)(64,71,88)(65,138,135)(66,139,136)(67,140,133)(68,137,134)(77,114,84)(78,115,81)(79,116,82)(80,113,83)(97,128,104)(98,125,101)(99,126,102)(100,127,103), (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)(5,101)(6,104)(7,103)(8,102)(9,98)(10,97)(11,100)(12,99)(13,53)(14,56)(15,55)(16,54)(17,46)(18,45)(19,48)(20,47)(21,37)(22,40)(23,39)(24,38)(25,63)(26,62)(27,61)(28,64)(29,74)(30,73)(31,76)(32,75)(33,86)(34,85)(35,88)(36,87)(41,132)(42,131)(43,130)(44,129)(49,133)(50,136)(51,135)(52,134)(57,58)(59,60)(65,109)(66,112)(67,111)(68,110)(69,118)(70,117)(71,120)(72,119)(77,78)(79,80)(81,114)(82,113)(83,116)(84,115)(89,90)(91,92)(93,123)(94,122)(95,121)(96,124)(105,137)(106,140)(107,139)(108,138)(125,141)(126,144)(127,143)(128,142), (1,90,3,92)(2,91,4,89)(5,30,7,32)(6,31,8,29)(9,132,11,130)(10,129,12,131)(13,124,15,122)(14,121,16,123)(17,113,19,115)(18,114,20,116)(21,141,23,143)(22,142,24,144)(25,139,27,137)(26,140,28,138)(33,67,35,65)(34,68,36,66)(37,125,39,127)(38,126,40,128)(41,100,43,98)(42,97,44,99)(45,81,47,83)(46,82,48,84)(49,71,51,69)(50,72,52,70)(53,96,55,94)(54,93,56,95)(57,79,59,77)(58,80,60,78)(61,105,63,107)(62,106,64,108)(73,103,75,101)(74,104,76,102)(85,110,87,112)(86,111,88,109)(117,136,119,134)(118,133,120,135), (1,78,3,80)(2,79,4,77)(5,52,7,50)(6,49,8,51)(9,110,11,112)(10,111,12,109)(13,46,15,48)(14,47,16,45)(17,55,19,53)(18,56,20,54)(21,63,23,61)(22,64,24,62)(25,39,27,37)(26,40,28,38)(29,71,31,69)(30,72,32,70)(33,42,35,44)(34,43,36,41)(57,89,59,91)(58,90,60,92)(65,97,67,99)(66,98,68,100)(73,119,75,117)(74,120,76,118)(81,123,83,121)(82,124,84,122)(85,130,87,132)(86,131,88,129)(93,116,95,114)(94,113,96,115)(101,134,103,136)(102,135,104,133)(105,143,107,141)(106,144,108,142)(125,137,127,139)(126,138,128,140) );
G=PermutationGroup([(1,32,76),(2,29,73),(3,30,74),(4,31,75),(5,90,102),(6,91,103),(7,92,104),(8,89,101),(9,55,126),(10,56,127),(11,53,128),(12,54,125),(13,100,142),(14,97,143),(15,98,144),(16,99,141),(17,140,112),(18,137,109),(19,138,110),(20,139,111),(21,44,121),(22,41,122),(23,42,123),(24,43,124),(25,114,86),(26,115,87),(27,116,88),(28,113,85),(33,81,63),(34,82,64),(35,83,61),(36,84,62),(37,95,129),(38,96,130),(39,93,131),(40,94,132),(45,65,105),(46,66,106),(47,67,107),(48,68,108),(49,57,136),(50,58,133),(51,59,134),(52,60,135),(69,117,77),(70,118,78),(71,119,79),(72,120,80)], [(1,96,121),(2,93,122),(3,94,123),(4,95,124),(5,141,9),(6,142,10),(7,143,11),(8,144,12),(13,56,91),(14,53,92),(15,54,89),(16,55,90),(17,58,47),(18,59,48),(19,60,45),(20,57,46),(21,32,130),(22,29,131),(23,30,132),(24,31,129),(25,36,117),(26,33,118),(27,34,119),(28,35,120),(37,43,75),(38,44,76),(39,41,73),(40,42,74),(49,106,111),(50,107,112),(51,108,109),(52,105,110),(61,72,85),(62,69,86),(63,70,87),(64,71,88),(65,138,135),(66,139,136),(67,140,133),(68,137,134),(77,114,84),(78,115,81),(79,116,82),(80,113,83),(97,128,104),(98,125,101),(99,126,102),(100,127,103)], [(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),(5,101),(6,104),(7,103),(8,102),(9,98),(10,97),(11,100),(12,99),(13,53),(14,56),(15,55),(16,54),(17,46),(18,45),(19,48),(20,47),(21,37),(22,40),(23,39),(24,38),(25,63),(26,62),(27,61),(28,64),(29,74),(30,73),(31,76),(32,75),(33,86),(34,85),(35,88),(36,87),(41,132),(42,131),(43,130),(44,129),(49,133),(50,136),(51,135),(52,134),(57,58),(59,60),(65,109),(66,112),(67,111),(68,110),(69,118),(70,117),(71,120),(72,119),(77,78),(79,80),(81,114),(82,113),(83,116),(84,115),(89,90),(91,92),(93,123),(94,122),(95,121),(96,124),(105,137),(106,140),(107,139),(108,138),(125,141),(126,144),(127,143),(128,142)], [(1,90,3,92),(2,91,4,89),(5,30,7,32),(6,31,8,29),(9,132,11,130),(10,129,12,131),(13,124,15,122),(14,121,16,123),(17,113,19,115),(18,114,20,116),(21,141,23,143),(22,142,24,144),(25,139,27,137),(26,140,28,138),(33,67,35,65),(34,68,36,66),(37,125,39,127),(38,126,40,128),(41,100,43,98),(42,97,44,99),(45,81,47,83),(46,82,48,84),(49,71,51,69),(50,72,52,70),(53,96,55,94),(54,93,56,95),(57,79,59,77),(58,80,60,78),(61,105,63,107),(62,106,64,108),(73,103,75,101),(74,104,76,102),(85,110,87,112),(86,111,88,109),(117,136,119,134),(118,133,120,135)], [(1,78,3,80),(2,79,4,77),(5,52,7,50),(6,49,8,51),(9,110,11,112),(10,111,12,109),(13,46,15,48),(14,47,16,45),(17,55,19,53),(18,56,20,54),(21,63,23,61),(22,64,24,62),(25,39,27,37),(26,40,28,38),(29,71,31,69),(30,72,32,70),(33,42,35,44),(34,43,36,41),(57,89,59,91),(58,90,60,92),(65,97,67,99),(66,98,68,100),(73,119,75,117),(74,120,76,118),(81,123,83,121),(82,124,84,122),(85,130,87,132),(86,131,88,129),(93,116,95,114),(94,113,96,115),(101,134,103,136),(102,135,104,133),(105,143,107,141),(106,144,108,142),(125,137,127,139),(126,138,128,140)])
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