direct product, metabelian, supersoluble, monomial
Aliases: C22×C32⋊7D4, C62⋊26D4, C62⋊9C23, C24⋊6(C3⋊S3), (C23×C6)⋊10S3, (C22×C6)⋊16D6, C6.68(S3×C23), (C3×C6).67C24, C3⋊Dic3⋊8C23, (C22×C62)⋊6C2, C32⋊14(C22×D4), (C2×C62)⋊16C22, C6⋊4(C2×C3⋊D4), (C3×C6)⋊13(C2×D4), C23⋊5(C2×C3⋊S3), (C23×C3⋊S3)⋊8C2, (C2×C3⋊S3)⋊8C23, C3⋊4(C22×C3⋊D4), (C2×C6)⋊18(C3⋊D4), (C2×C6)⋊11(C22×S3), C2.15(C23×C3⋊S3), C22⋊3(C22×C3⋊S3), (C22×C3⋊S3)⋊17C22, (C22×C3⋊Dic3)⋊16C2, (C2×C3⋊Dic3)⋊29C22, SmallGroup(288,1017)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C32⋊7D4
G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf=c-1, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 2404 in 708 conjugacy classes, 213 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C23, C32, Dic3, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C24, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C22×D4, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C22×Dic3, C2×C3⋊D4, S3×C23, C23×C6, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C2×C62, C2×C62, C22×C3⋊D4, C22×C3⋊Dic3, C2×C32⋊7D4, C23×C3⋊S3, C22×C62, C22×C32⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊S3, C3⋊D4, C22×S3, C22×D4, C2×C3⋊S3, C2×C3⋊D4, S3×C23, C32⋊7D4, C22×C3⋊S3, C22×C3⋊D4, C2×C32⋊7D4, C23×C3⋊S3, C22×C32⋊7D4
►On 144 points
Generators in S144
(1 78)(2 79)(3 80)(4 77)(5 94)(6 95)(7 96)(8 93)(9 39)(10 40)(11 37)(12 38)(13 136)(14 133)(15 134)(16 135)(17 120)(18 117)(19 118)(20 119)(21 69)(22 70)(23 71)(24 72)(25 108)(26 105)(27 106)(28 107)(29 54)(30 55)(31 56)(32 53)(33 81)(34 82)(35 83)(36 84)(41 51)(42 52)(43 49)(44 50)(45 101)(46 102)(47 103)(48 104)(57 127)(58 128)(59 125)(60 126)(61 98)(62 99)(63 100)(64 97)(65 75)(66 76)(67 73)(68 74)(85 130)(86 131)(87 132)(88 129)(89 111)(90 112)(91 109)(92 110)(113 121)(114 122)(115 123)(116 124)(137 142)(138 143)(139 144)(140 141)
(1 99)(2 100)(3 97)(4 98)(5 82)(6 83)(7 84)(8 81)(9 55)(10 56)(11 53)(12 54)(13 76)(14 73)(15 74)(16 75)(17 103)(18 104)(19 101)(20 102)(21 105)(22 106)(23 107)(24 108)(25 72)(26 69)(27 70)(28 71)(29 38)(30 39)(31 40)(32 37)(33 93)(34 94)(35 95)(36 96)(41 122)(42 123)(43 124)(44 121)(45 118)(46 119)(47 120)(48 117)(49 116)(50 113)(51 114)(52 115)(57 137)(58 138)(59 139)(60 140)(61 77)(62 78)(63 79)(64 80)(65 135)(66 136)(67 133)(68 134)(85 110)(86 111)(87 112)(88 109)(89 131)(90 132)(91 129)(92 130)(125 144)(126 141)(127 142)(128 143)
(1 73 19)(2 20 74)(3 75 17)(4 18 76)(5 138 110)(6 111 139)(7 140 112)(8 109 137)(9 25 42)(10 43 26)(11 27 44)(12 41 28)(13 98 104)(14 101 99)(15 100 102)(16 103 97)(21 31 116)(22 113 32)(23 29 114)(24 115 30)(33 129 127)(34 128 130)(35 131 125)(36 126 132)(37 106 50)(38 51 107)(39 108 52)(40 49 105)(45 62 133)(46 134 63)(47 64 135)(48 136 61)(53 70 121)(54 122 71)(55 72 123)(56 124 69)(57 81 88)(58 85 82)(59 83 86)(60 87 84)(65 120 80)(66 77 117)(67 118 78)(68 79 119)(89 144 95)(90 96 141)(91 142 93)(92 94 143)
(1 123 36)(2 33 124)(3 121 34)(4 35 122)(5 64 50)(6 51 61)(7 62 52)(8 49 63)(9 141 14)(10 15 142)(11 143 16)(12 13 144)(17 70 130)(18 131 71)(19 72 132)(20 129 69)(21 119 88)(22 85 120)(23 117 86)(24 87 118)(25 90 101)(26 102 91)(27 92 103)(28 104 89)(29 66 59)(30 60 67)(31 68 57)(32 58 65)(37 138 135)(38 136 139)(39 140 133)(40 134 137)(41 98 95)(42 96 99)(43 100 93)(44 94 97)(45 108 112)(46 109 105)(47 106 110)(48 111 107)(53 128 75)(54 76 125)(55 126 73)(56 74 127)(77 83 114)(78 115 84)(79 81 116)(80 113 82)
(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 62)(2 61)(3 64)(4 63)(5 121)(6 124)(7 123)(8 122)(9 87)(10 86)(11 85)(12 88)(13 119)(14 118)(15 117)(16 120)(17 135)(18 134)(19 133)(20 136)(21 144)(22 143)(23 142)(24 141)(25 60)(26 59)(27 58)(28 57)(29 91)(30 90)(31 89)(32 92)(33 51)(34 50)(35 49)(36 52)(37 130)(38 129)(39 132)(40 131)(41 81)(42 84)(43 83)(44 82)(45 73)(46 76)(47 75)(48 74)(53 110)(54 109)(55 112)(56 111)(65 103)(66 102)(67 101)(68 104)(69 139)(70 138)(71 137)(72 140)(77 100)(78 99)(79 98)(80 97)(93 114)(94 113)(95 116)(96 115)(105 125)(106 128)(107 127)(108 126)
G:=sub<Sym(144)| (1,78)(2,79)(3,80)(4,77)(5,94)(6,95)(7,96)(8,93)(9,39)(10,40)(11,37)(12,38)(13,136)(14,133)(15,134)(16,135)(17,120)(18,117)(19,118)(20,119)(21,69)(22,70)(23,71)(24,72)(25,108)(26,105)(27,106)(28,107)(29,54)(30,55)(31,56)(32,53)(33,81)(34,82)(35,83)(36,84)(41,51)(42,52)(43,49)(44,50)(45,101)(46,102)(47,103)(48,104)(57,127)(58,128)(59,125)(60,126)(61,98)(62,99)(63,100)(64,97)(65,75)(66,76)(67,73)(68,74)(85,130)(86,131)(87,132)(88,129)(89,111)(90,112)(91,109)(92,110)(113,121)(114,122)(115,123)(116,124)(137,142)(138,143)(139,144)(140,141), (1,99)(2,100)(3,97)(4,98)(5,82)(6,83)(7,84)(8,81)(9,55)(10,56)(11,53)(12,54)(13,76)(14,73)(15,74)(16,75)(17,103)(18,104)(19,101)(20,102)(21,105)(22,106)(23,107)(24,108)(25,72)(26,69)(27,70)(28,71)(29,38)(30,39)(31,40)(32,37)(33,93)(34,94)(35,95)(36,96)(41,122)(42,123)(43,124)(44,121)(45,118)(46,119)(47,120)(48,117)(49,116)(50,113)(51,114)(52,115)(57,137)(58,138)(59,139)(60,140)(61,77)(62,78)(63,79)(64,80)(65,135)(66,136)(67,133)(68,134)(85,110)(86,111)(87,112)(88,109)(89,131)(90,132)(91,129)(92,130)(125,144)(126,141)(127,142)(128,143), (1,73,19)(2,20,74)(3,75,17)(4,18,76)(5,138,110)(6,111,139)(7,140,112)(8,109,137)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(13,98,104)(14,101,99)(15,100,102)(16,103,97)(21,31,116)(22,113,32)(23,29,114)(24,115,30)(33,129,127)(34,128,130)(35,131,125)(36,126,132)(37,106,50)(38,51,107)(39,108,52)(40,49,105)(45,62,133)(46,134,63)(47,64,135)(48,136,61)(53,70,121)(54,122,71)(55,72,123)(56,124,69)(57,81,88)(58,85,82)(59,83,86)(60,87,84)(65,120,80)(66,77,117)(67,118,78)(68,79,119)(89,144,95)(90,96,141)(91,142,93)(92,94,143), (1,123,36)(2,33,124)(3,121,34)(4,35,122)(5,64,50)(6,51,61)(7,62,52)(8,49,63)(9,141,14)(10,15,142)(11,143,16)(12,13,144)(17,70,130)(18,131,71)(19,72,132)(20,129,69)(21,119,88)(22,85,120)(23,117,86)(24,87,118)(25,90,101)(26,102,91)(27,92,103)(28,104,89)(29,66,59)(30,60,67)(31,68,57)(32,58,65)(37,138,135)(38,136,139)(39,140,133)(40,134,137)(41,98,95)(42,96,99)(43,100,93)(44,94,97)(45,108,112)(46,109,105)(47,106,110)(48,111,107)(53,128,75)(54,76,125)(55,126,73)(56,74,127)(77,83,114)(78,115,84)(79,81,116)(80,113,82), (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), (1,62)(2,61)(3,64)(4,63)(5,121)(6,124)(7,123)(8,122)(9,87)(10,86)(11,85)(12,88)(13,119)(14,118)(15,117)(16,120)(17,135)(18,134)(19,133)(20,136)(21,144)(22,143)(23,142)(24,141)(25,60)(26,59)(27,58)(28,57)(29,91)(30,90)(31,89)(32,92)(33,51)(34,50)(35,49)(36,52)(37,130)(38,129)(39,132)(40,131)(41,81)(42,84)(43,83)(44,82)(45,73)(46,76)(47,75)(48,74)(53,110)(54,109)(55,112)(56,111)(65,103)(66,102)(67,101)(68,104)(69,139)(70,138)(71,137)(72,140)(77,100)(78,99)(79,98)(80,97)(93,114)(94,113)(95,116)(96,115)(105,125)(106,128)(107,127)(108,126)>;
G:=Group( (1,78)(2,79)(3,80)(4,77)(5,94)(6,95)(7,96)(8,93)(9,39)(10,40)(11,37)(12,38)(13,136)(14,133)(15,134)(16,135)(17,120)(18,117)(19,118)(20,119)(21,69)(22,70)(23,71)(24,72)(25,108)(26,105)(27,106)(28,107)(29,54)(30,55)(31,56)(32,53)(33,81)(34,82)(35,83)(36,84)(41,51)(42,52)(43,49)(44,50)(45,101)(46,102)(47,103)(48,104)(57,127)(58,128)(59,125)(60,126)(61,98)(62,99)(63,100)(64,97)(65,75)(66,76)(67,73)(68,74)(85,130)(86,131)(87,132)(88,129)(89,111)(90,112)(91,109)(92,110)(113,121)(114,122)(115,123)(116,124)(137,142)(138,143)(139,144)(140,141), (1,99)(2,100)(3,97)(4,98)(5,82)(6,83)(7,84)(8,81)(9,55)(10,56)(11,53)(12,54)(13,76)(14,73)(15,74)(16,75)(17,103)(18,104)(19,101)(20,102)(21,105)(22,106)(23,107)(24,108)(25,72)(26,69)(27,70)(28,71)(29,38)(30,39)(31,40)(32,37)(33,93)(34,94)(35,95)(36,96)(41,122)(42,123)(43,124)(44,121)(45,118)(46,119)(47,120)(48,117)(49,116)(50,113)(51,114)(52,115)(57,137)(58,138)(59,139)(60,140)(61,77)(62,78)(63,79)(64,80)(65,135)(66,136)(67,133)(68,134)(85,110)(86,111)(87,112)(88,109)(89,131)(90,132)(91,129)(92,130)(125,144)(126,141)(127,142)(128,143), (1,73,19)(2,20,74)(3,75,17)(4,18,76)(5,138,110)(6,111,139)(7,140,112)(8,109,137)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(13,98,104)(14,101,99)(15,100,102)(16,103,97)(21,31,116)(22,113,32)(23,29,114)(24,115,30)(33,129,127)(34,128,130)(35,131,125)(36,126,132)(37,106,50)(38,51,107)(39,108,52)(40,49,105)(45,62,133)(46,134,63)(47,64,135)(48,136,61)(53,70,121)(54,122,71)(55,72,123)(56,124,69)(57,81,88)(58,85,82)(59,83,86)(60,87,84)(65,120,80)(66,77,117)(67,118,78)(68,79,119)(89,144,95)(90,96,141)(91,142,93)(92,94,143), (1,123,36)(2,33,124)(3,121,34)(4,35,122)(5,64,50)(6,51,61)(7,62,52)(8,49,63)(9,141,14)(10,15,142)(11,143,16)(12,13,144)(17,70,130)(18,131,71)(19,72,132)(20,129,69)(21,119,88)(22,85,120)(23,117,86)(24,87,118)(25,90,101)(26,102,91)(27,92,103)(28,104,89)(29,66,59)(30,60,67)(31,68,57)(32,58,65)(37,138,135)(38,136,139)(39,140,133)(40,134,137)(41,98,95)(42,96,99)(43,100,93)(44,94,97)(45,108,112)(46,109,105)(47,106,110)(48,111,107)(53,128,75)(54,76,125)(55,126,73)(56,74,127)(77,83,114)(78,115,84)(79,81,116)(80,113,82), (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), (1,62)(2,61)(3,64)(4,63)(5,121)(6,124)(7,123)(8,122)(9,87)(10,86)(11,85)(12,88)(13,119)(14,118)(15,117)(16,120)(17,135)(18,134)(19,133)(20,136)(21,144)(22,143)(23,142)(24,141)(25,60)(26,59)(27,58)(28,57)(29,91)(30,90)(31,89)(32,92)(33,51)(34,50)(35,49)(36,52)(37,130)(38,129)(39,132)(40,131)(41,81)(42,84)(43,83)(44,82)(45,73)(46,76)(47,75)(48,74)(53,110)(54,109)(55,112)(56,111)(65,103)(66,102)(67,101)(68,104)(69,139)(70,138)(71,137)(72,140)(77,100)(78,99)(79,98)(80,97)(93,114)(94,113)(95,116)(96,115)(105,125)(106,128)(107,127)(108,126) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,77),(5,94),(6,95),(7,96),(8,93),(9,39),(10,40),(11,37),(12,38),(13,136),(14,133),(15,134),(16,135),(17,120),(18,117),(19,118),(20,119),(21,69),(22,70),(23,71),(24,72),(25,108),(26,105),(27,106),(28,107),(29,54),(30,55),(31,56),(32,53),(33,81),(34,82),(35,83),(36,84),(41,51),(42,52),(43,49),(44,50),(45,101),(46,102),(47,103),(48,104),(57,127),(58,128),(59,125),(60,126),(61,98),(62,99),(63,100),(64,97),(65,75),(66,76),(67,73),(68,74),(85,130),(86,131),(87,132),(88,129),(89,111),(90,112),(91,109),(92,110),(113,121),(114,122),(115,123),(116,124),(137,142),(138,143),(139,144),(140,141)], [(1,99),(2,100),(3,97),(4,98),(5,82),(6,83),(7,84),(8,81),(9,55),(10,56),(11,53),(12,54),(13,76),(14,73),(15,74),(16,75),(17,103),(18,104),(19,101),(20,102),(21,105),(22,106),(23,107),(24,108),(25,72),(26,69),(27,70),(28,71),(29,38),(30,39),(31,40),(32,37),(33,93),(34,94),(35,95),(36,96),(41,122),(42,123),(43,124),(44,121),(45,118),(46,119),(47,120),(48,117),(49,116),(50,113),(51,114),(52,115),(57,137),(58,138),(59,139),(60,140),(61,77),(62,78),(63,79),(64,80),(65,135),(66,136),(67,133),(68,134),(85,110),(86,111),(87,112),(88,109),(89,131),(90,132),(91,129),(92,130),(125,144),(126,141),(127,142),(128,143)], [(1,73,19),(2,20,74),(3,75,17),(4,18,76),(5,138,110),(6,111,139),(7,140,112),(8,109,137),(9,25,42),(10,43,26),(11,27,44),(12,41,28),(13,98,104),(14,101,99),(15,100,102),(16,103,97),(21,31,116),(22,113,32),(23,29,114),(24,115,30),(33,129,127),(34,128,130),(35,131,125),(36,126,132),(37,106,50),(38,51,107),(39,108,52),(40,49,105),(45,62,133),(46,134,63),(47,64,135),(48,136,61),(53,70,121),(54,122,71),(55,72,123),(56,124,69),(57,81,88),(58,85,82),(59,83,86),(60,87,84),(65,120,80),(66,77,117),(67,118,78),(68,79,119),(89,144,95),(90,96,141),(91,142,93),(92,94,143)], [(1,123,36),(2,33,124),(3,121,34),(4,35,122),(5,64,50),(6,51,61),(7,62,52),(8,49,63),(9,141,14),(10,15,142),(11,143,16),(12,13,144),(17,70,130),(18,131,71),(19,72,132),(20,129,69),(21,119,88),(22,85,120),(23,117,86),(24,87,118),(25,90,101),(26,102,91),(27,92,103),(28,104,89),(29,66,59),(30,60,67),(31,68,57),(32,58,65),(37,138,135),(38,136,139),(39,140,133),(40,134,137),(41,98,95),(42,96,99),(43,100,93),(44,94,97),(45,108,112),(46,109,105),(47,106,110),(48,111,107),(53,128,75),(54,76,125),(55,126,73),(56,74,127),(77,83,114),(78,115,84),(79,81,116),(80,113,82)], [(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)], [(1,62),(2,61),(3,64),(4,63),(5,121),(6,124),(7,123),(8,122),(9,87),(10,86),(11,85),(12,88),(13,119),(14,118),(15,117),(16,120),(17,135),(18,134),(19,133),(20,136),(21,144),(22,143),(23,142),(24,141),(25,60),(26,59),(27,58),(28,57),(29,91),(30,90),(31,89),(32,92),(33,51),(34,50),(35,49),(36,52),(37,130),(38,129),(39,132),(40,131),(41,81),(42,84),(43,83),(44,82),(45,73),(46,76),(47,75),(48,74),(53,110),(54,109),(55,112),(56,111),(65,103),(66,102),(67,101),(68,104),(69,139),(70,138),(71,137),(72,140),(77,100),(78,99),(79,98),(80,97),(93,114),(94,113),(95,116),(96,115),(105,125),(106,128),(107,127),(108,126)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 |
| kernel | C22×C32⋊7D4 | C22×C3⋊Dic3 | C2×C32⋊7D4 | C23×C3⋊S3 | C22×C62 | C23×C6 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 1 | 12 | 1 | 1 | 4 | 4 | 28 | 32 |
Matrix representation of C22×C32⋊7D4 ►in GL5(𝔽13)
| 12 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 11 | 2 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,1,12],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,1,12],[12,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,11,11,0,0,0,9,2],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,1] >;
C22×C32⋊7D4 in GAP, Magma, Sage, TeX
C_2^2\times C_3^2\rtimes_7D_4
% in TeX
G:=Group("C2^2xC3^2:7D4"); // GroupNames label
G:=SmallGroup(288,1017);
// by ID
G=gap.SmallGroup(288,1017);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=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,c*d=d*c,e*c*e^-1=f*c*f=c^-1,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations