direct product, non-abelian, supersoluble, monomial
Aliases: C22×He3⋊3C4, C62⋊8Dic3, C62.48D6, He3⋊8(C22×C4), (C2×C62).13S3, (C22×He3)⋊7C4, (C23×He3).5C2, (C2×He3).37C23, C32⋊3(C22×Dic3), C23.3(He3⋊C2), (C22×He3).35C22, (C2×He3)⋊7(C2×C4), (C3×C6)⋊3(C2×Dic3), C6.50(C22×C3⋊S3), C6.19(C2×C3⋊Dic3), (C3×C6).47(C22×S3), C3.2(C22×C3⋊Dic3), (C22×C6).18(C3⋊S3), (C2×C6).14(C3⋊Dic3), C2.2(C22×He3⋊C2), C22.11(C2×He3⋊C2), (C2×C6).59(C2×C3⋊S3), SmallGroup(432,398)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C22×He3⋊3C4 |
Generators and relations for C22×He3⋊3C4
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf-1=c-1, de=ed, df=fd, fef-1=e-1 >
Subgroups: 777 in 297 conjugacy classes, 123 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, He3, C3×Dic3, C62, C22×Dic3, C22×C12, C2×He3, C2×He3, C6×Dic3, C2×C62, He3⋊3C4, C22×He3, Dic3×C2×C6, C2×He3⋊3C4, C23×He3, C22×He3⋊3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C3⋊S3, C2×Dic3, C22×S3, C3⋊Dic3, C2×C3⋊S3, C22×Dic3, He3⋊C2, C2×C3⋊Dic3, C22×C3⋊S3, He3⋊3C4, C2×He3⋊C2, C22×C3⋊Dic3, C2×He3⋊3C4, C22×He3⋊C2, C22×He3⋊3C4
►On 144 points
Generators in S144
(1 116)(2 113)(3 114)(4 115)(5 106)(6 107)(7 108)(8 105)(9 35)(10 36)(11 33)(12 34)(13 112)(14 109)(15 110)(16 111)(17 117)(18 118)(19 119)(20 120)(21 129)(22 130)(23 131)(24 132)(25 142)(26 143)(27 144)(28 141)(29 137)(30 138)(31 139)(32 140)(37 57)(38 58)(39 59)(40 60)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(61 102)(62 103)(63 104)(64 101)(65 96)(66 93)(67 94)(68 95)(69 100)(70 97)(71 98)(72 99)(89 126)(90 127)(91 128)(92 125)(121 135)(122 136)(123 133)(124 134)
(1 77)(2 78)(3 79)(4 80)(5 58)(6 59)(7 60)(8 57)(9 72)(10 69)(11 70)(12 71)(13 73)(14 74)(15 75)(16 76)(17 82)(18 83)(19 84)(20 81)(21 90)(22 91)(23 92)(24 89)(25 103)(26 104)(27 101)(28 102)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 105)(38 106)(39 107)(40 108)(41 112)(42 109)(43 110)(44 111)(45 116)(46 113)(47 114)(48 115)(49 120)(50 117)(51 118)(52 119)(53 124)(54 121)(55 122)(56 123)(61 141)(62 142)(63 143)(64 144)(65 140)(66 137)(67 138)(68 139)(85 134)(86 135)(87 136)(88 133)(125 131)(126 132)(127 129)(128 130)
(1 68 59)(2 60 65)(3 66 57)(4 58 67)(5 138 80)(6 77 139)(7 140 78)(8 79 137)(9 19 126)(10 127 20)(11 17 128)(12 125 18)(13 86 61)(14 62 87)(15 88 63)(16 64 85)(21 49 100)(22 97 50)(23 51 98)(24 99 52)(25 122 42)(26 43 123)(27 124 44)(28 41 121)(29 105 47)(30 48 106)(31 107 45)(32 46 108)(33 117 91)(34 92 118)(35 119 89)(36 90 120)(37 114 93)(38 94 115)(39 116 95)(40 96 113)(53 111 101)(54 102 112)(55 109 103)(56 104 110)(69 129 81)(70 82 130)(71 131 83)(72 84 132)(73 135 141)(74 142 136)(75 133 143)(76 144 134)
(1 73 19)(2 74 20)(3 75 17)(4 76 18)(5 64 71)(6 61 72)(7 62 69)(8 63 70)(9 59 141)(10 60 142)(11 57 143)(12 58 144)(13 84 77)(14 81 78)(15 82 79)(16 83 80)(21 32 55)(22 29 56)(23 30 53)(24 31 54)(25 36 40)(26 33 37)(27 34 38)(28 35 39)(41 119 116)(42 120 113)(43 117 114)(44 118 115)(45 112 52)(46 109 49)(47 110 50)(48 111 51)(65 136 127)(66 133 128)(67 134 125)(68 135 126)(85 131 138)(86 132 139)(87 129 140)(88 130 137)(89 95 121)(90 96 122)(91 93 123)(92 94 124)(97 105 104)(98 106 101)(99 107 102)(100 108 103)
(1 68 141)(2 142 65)(3 66 143)(4 144 67)(5 131 83)(6 84 132)(7 129 81)(8 82 130)(9 73 135)(10 136 74)(11 75 133)(12 134 76)(13 86 72)(14 69 87)(15 88 70)(16 71 85)(17 128 57)(18 58 125)(19 126 59)(20 60 127)(21 49 108)(22 105 50)(23 51 106)(24 107 52)(25 96 113)(26 114 93)(27 94 115)(28 116 95)(29 104 47)(30 48 101)(31 102 45)(32 46 103)(33 43 123)(34 124 44)(35 41 121)(36 122 42)(37 117 91)(38 92 118)(39 119 89)(40 90 120)(53 111 98)(54 99 112)(55 109 100)(56 97 110)(61 77 139)(62 140 78)(63 79 137)(64 138 80)
(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)
G:=sub<Sym(144)| (1,116)(2,113)(3,114)(4,115)(5,106)(6,107)(7,108)(8,105)(9,35)(10,36)(11,33)(12,34)(13,112)(14,109)(15,110)(16,111)(17,117)(18,118)(19,119)(20,120)(21,129)(22,130)(23,131)(24,132)(25,142)(26,143)(27,144)(28,141)(29,137)(30,138)(31,139)(32,140)(37,57)(38,58)(39,59)(40,60)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(61,102)(62,103)(63,104)(64,101)(65,96)(66,93)(67,94)(68,95)(69,100)(70,97)(71,98)(72,99)(89,126)(90,127)(91,128)(92,125)(121,135)(122,136)(123,133)(124,134), (1,77)(2,78)(3,79)(4,80)(5,58)(6,59)(7,60)(8,57)(9,72)(10,69)(11,70)(12,71)(13,73)(14,74)(15,75)(16,76)(17,82)(18,83)(19,84)(20,81)(21,90)(22,91)(23,92)(24,89)(25,103)(26,104)(27,101)(28,102)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,105)(38,106)(39,107)(40,108)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,120)(50,117)(51,118)(52,119)(53,124)(54,121)(55,122)(56,123)(61,141)(62,142)(63,143)(64,144)(65,140)(66,137)(67,138)(68,139)(85,134)(86,135)(87,136)(88,133)(125,131)(126,132)(127,129)(128,130), (1,68,59)(2,60,65)(3,66,57)(4,58,67)(5,138,80)(6,77,139)(7,140,78)(8,79,137)(9,19,126)(10,127,20)(11,17,128)(12,125,18)(13,86,61)(14,62,87)(15,88,63)(16,64,85)(21,49,100)(22,97,50)(23,51,98)(24,99,52)(25,122,42)(26,43,123)(27,124,44)(28,41,121)(29,105,47)(30,48,106)(31,107,45)(32,46,108)(33,117,91)(34,92,118)(35,119,89)(36,90,120)(37,114,93)(38,94,115)(39,116,95)(40,96,113)(53,111,101)(54,102,112)(55,109,103)(56,104,110)(69,129,81)(70,82,130)(71,131,83)(72,84,132)(73,135,141)(74,142,136)(75,133,143)(76,144,134), (1,73,19)(2,74,20)(3,75,17)(4,76,18)(5,64,71)(6,61,72)(7,62,69)(8,63,70)(9,59,141)(10,60,142)(11,57,143)(12,58,144)(13,84,77)(14,81,78)(15,82,79)(16,83,80)(21,32,55)(22,29,56)(23,30,53)(24,31,54)(25,36,40)(26,33,37)(27,34,38)(28,35,39)(41,119,116)(42,120,113)(43,117,114)(44,118,115)(45,112,52)(46,109,49)(47,110,50)(48,111,51)(65,136,127)(66,133,128)(67,134,125)(68,135,126)(85,131,138)(86,132,139)(87,129,140)(88,130,137)(89,95,121)(90,96,122)(91,93,123)(92,94,124)(97,105,104)(98,106,101)(99,107,102)(100,108,103), (1,68,141)(2,142,65)(3,66,143)(4,144,67)(5,131,83)(6,84,132)(7,129,81)(8,82,130)(9,73,135)(10,136,74)(11,75,133)(12,134,76)(13,86,72)(14,69,87)(15,88,70)(16,71,85)(17,128,57)(18,58,125)(19,126,59)(20,60,127)(21,49,108)(22,105,50)(23,51,106)(24,107,52)(25,96,113)(26,114,93)(27,94,115)(28,116,95)(29,104,47)(30,48,101)(31,102,45)(32,46,103)(33,43,123)(34,124,44)(35,41,121)(36,122,42)(37,117,91)(38,92,118)(39,119,89)(40,90,120)(53,111,98)(54,99,112)(55,109,100)(56,97,110)(61,77,139)(62,140,78)(63,79,137)(64,138,80), (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)>;
G:=Group( (1,116)(2,113)(3,114)(4,115)(5,106)(6,107)(7,108)(8,105)(9,35)(10,36)(11,33)(12,34)(13,112)(14,109)(15,110)(16,111)(17,117)(18,118)(19,119)(20,120)(21,129)(22,130)(23,131)(24,132)(25,142)(26,143)(27,144)(28,141)(29,137)(30,138)(31,139)(32,140)(37,57)(38,58)(39,59)(40,60)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(61,102)(62,103)(63,104)(64,101)(65,96)(66,93)(67,94)(68,95)(69,100)(70,97)(71,98)(72,99)(89,126)(90,127)(91,128)(92,125)(121,135)(122,136)(123,133)(124,134), (1,77)(2,78)(3,79)(4,80)(5,58)(6,59)(7,60)(8,57)(9,72)(10,69)(11,70)(12,71)(13,73)(14,74)(15,75)(16,76)(17,82)(18,83)(19,84)(20,81)(21,90)(22,91)(23,92)(24,89)(25,103)(26,104)(27,101)(28,102)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,105)(38,106)(39,107)(40,108)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,120)(50,117)(51,118)(52,119)(53,124)(54,121)(55,122)(56,123)(61,141)(62,142)(63,143)(64,144)(65,140)(66,137)(67,138)(68,139)(85,134)(86,135)(87,136)(88,133)(125,131)(126,132)(127,129)(128,130), (1,68,59)(2,60,65)(3,66,57)(4,58,67)(5,138,80)(6,77,139)(7,140,78)(8,79,137)(9,19,126)(10,127,20)(11,17,128)(12,125,18)(13,86,61)(14,62,87)(15,88,63)(16,64,85)(21,49,100)(22,97,50)(23,51,98)(24,99,52)(25,122,42)(26,43,123)(27,124,44)(28,41,121)(29,105,47)(30,48,106)(31,107,45)(32,46,108)(33,117,91)(34,92,118)(35,119,89)(36,90,120)(37,114,93)(38,94,115)(39,116,95)(40,96,113)(53,111,101)(54,102,112)(55,109,103)(56,104,110)(69,129,81)(70,82,130)(71,131,83)(72,84,132)(73,135,141)(74,142,136)(75,133,143)(76,144,134), (1,73,19)(2,74,20)(3,75,17)(4,76,18)(5,64,71)(6,61,72)(7,62,69)(8,63,70)(9,59,141)(10,60,142)(11,57,143)(12,58,144)(13,84,77)(14,81,78)(15,82,79)(16,83,80)(21,32,55)(22,29,56)(23,30,53)(24,31,54)(25,36,40)(26,33,37)(27,34,38)(28,35,39)(41,119,116)(42,120,113)(43,117,114)(44,118,115)(45,112,52)(46,109,49)(47,110,50)(48,111,51)(65,136,127)(66,133,128)(67,134,125)(68,135,126)(85,131,138)(86,132,139)(87,129,140)(88,130,137)(89,95,121)(90,96,122)(91,93,123)(92,94,124)(97,105,104)(98,106,101)(99,107,102)(100,108,103), (1,68,141)(2,142,65)(3,66,143)(4,144,67)(5,131,83)(6,84,132)(7,129,81)(8,82,130)(9,73,135)(10,136,74)(11,75,133)(12,134,76)(13,86,72)(14,69,87)(15,88,70)(16,71,85)(17,128,57)(18,58,125)(19,126,59)(20,60,127)(21,49,108)(22,105,50)(23,51,106)(24,107,52)(25,96,113)(26,114,93)(27,94,115)(28,116,95)(29,104,47)(30,48,101)(31,102,45)(32,46,103)(33,43,123)(34,124,44)(35,41,121)(36,122,42)(37,117,91)(38,92,118)(39,119,89)(40,90,120)(53,111,98)(54,99,112)(55,109,100)(56,97,110)(61,77,139)(62,140,78)(63,79,137)(64,138,80), (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) );
G=PermutationGroup([[(1,116),(2,113),(3,114),(4,115),(5,106),(6,107),(7,108),(8,105),(9,35),(10,36),(11,33),(12,34),(13,112),(14,109),(15,110),(16,111),(17,117),(18,118),(19,119),(20,120),(21,129),(22,130),(23,131),(24,132),(25,142),(26,143),(27,144),(28,141),(29,137),(30,138),(31,139),(32,140),(37,57),(38,58),(39,59),(40,60),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(61,102),(62,103),(63,104),(64,101),(65,96),(66,93),(67,94),(68,95),(69,100),(70,97),(71,98),(72,99),(89,126),(90,127),(91,128),(92,125),(121,135),(122,136),(123,133),(124,134)], [(1,77),(2,78),(3,79),(4,80),(5,58),(6,59),(7,60),(8,57),(9,72),(10,69),(11,70),(12,71),(13,73),(14,74),(15,75),(16,76),(17,82),(18,83),(19,84),(20,81),(21,90),(22,91),(23,92),(24,89),(25,103),(26,104),(27,101),(28,102),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,105),(38,106),(39,107),(40,108),(41,112),(42,109),(43,110),(44,111),(45,116),(46,113),(47,114),(48,115),(49,120),(50,117),(51,118),(52,119),(53,124),(54,121),(55,122),(56,123),(61,141),(62,142),(63,143),(64,144),(65,140),(66,137),(67,138),(68,139),(85,134),(86,135),(87,136),(88,133),(125,131),(126,132),(127,129),(128,130)], [(1,68,59),(2,60,65),(3,66,57),(4,58,67),(5,138,80),(6,77,139),(7,140,78),(8,79,137),(9,19,126),(10,127,20),(11,17,128),(12,125,18),(13,86,61),(14,62,87),(15,88,63),(16,64,85),(21,49,100),(22,97,50),(23,51,98),(24,99,52),(25,122,42),(26,43,123),(27,124,44),(28,41,121),(29,105,47),(30,48,106),(31,107,45),(32,46,108),(33,117,91),(34,92,118),(35,119,89),(36,90,120),(37,114,93),(38,94,115),(39,116,95),(40,96,113),(53,111,101),(54,102,112),(55,109,103),(56,104,110),(69,129,81),(70,82,130),(71,131,83),(72,84,132),(73,135,141),(74,142,136),(75,133,143),(76,144,134)], [(1,73,19),(2,74,20),(3,75,17),(4,76,18),(5,64,71),(6,61,72),(7,62,69),(8,63,70),(9,59,141),(10,60,142),(11,57,143),(12,58,144),(13,84,77),(14,81,78),(15,82,79),(16,83,80),(21,32,55),(22,29,56),(23,30,53),(24,31,54),(25,36,40),(26,33,37),(27,34,38),(28,35,39),(41,119,116),(42,120,113),(43,117,114),(44,118,115),(45,112,52),(46,109,49),(47,110,50),(48,111,51),(65,136,127),(66,133,128),(67,134,125),(68,135,126),(85,131,138),(86,132,139),(87,129,140),(88,130,137),(89,95,121),(90,96,122),(91,93,123),(92,94,124),(97,105,104),(98,106,101),(99,107,102),(100,108,103)], [(1,68,141),(2,142,65),(3,66,143),(4,144,67),(5,131,83),(6,84,132),(7,129,81),(8,82,130),(9,73,135),(10,136,74),(11,75,133),(12,134,76),(13,86,72),(14,69,87),(15,88,70),(16,71,85),(17,128,57),(18,58,125),(19,126,59),(20,60,127),(21,49,108),(22,105,50),(23,51,106),(24,107,52),(25,96,113),(26,114,93),(27,94,115),(28,116,95),(29,104,47),(30,48,101),(31,102,45),(32,46,103),(33,43,123),(34,124,44),(35,41,121),(36,122,42),(37,117,91),(38,92,118),(39,119,89),(40,90,120),(53,111,98),(54,99,112),(55,109,100),(56,97,110),(61,77,139),(62,140,78),(63,79,137),(64,138,80)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 3F | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AP | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | He3⋊C2 | He3⋊3C4 | C2×He3⋊C2 |
| kernel | C22×He3⋊3C4 | C2×He3⋊3C4 | C23×He3 | C22×He3 | C2×C62 | C62 | C62 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 4 | 16 | 12 | 4 | 16 | 12 |
Matrix representation of C22×He3⋊3C4 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 2 | 4 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 2 | 4 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 10 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,4,0,0,0,0,0,1,10],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9],[0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,3,0,2,0,0,0,0,1,0,4,0,0,0,0,10,9,10],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,4,1,4,0,0,0,0,9,0,10] >;
C22×He3⋊3C4 in GAP, Magma, Sage, TeX
C_2^2\times {\rm He}_3\rtimes_3C_4 % in TeX
G:=Group("C2^2xHe3:3C4"); // GroupNames label
G:=SmallGroup(432,398);
// by ID
G=gap.SmallGroup(432,398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^4=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=c*d^-1,f*c*f^-1=c^-1,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations