?
G = C32⋊92- (1+4) order 288 = 25·32
2nd semidirect product of C32 and 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial
Aliases: C32⋊92- (1+4), C62.155C23, C3⋊5(Q8○D12), (C3×D4).45D6, (C3×Q8).70D6, (C2×C12).173D6, C6.66(S3×C23), (C3×C6).65C24, C12.59D6⋊14C2, C12.D6⋊10C2, (C6×C12).172C22, C12.117(C22×S3), (C3×C12).159C23, C3⋊Dic3.53C23, C32⋊7D4.3C22, C12⋊S3.35C22, (D4×C32).31C22, (Q8×C32).34C22, C32⋊4Q8.37C22, (C3×C4○D4)⋊9S3, (Q8×C3⋊S3)⋊10C2, C4○D4⋊6(C3⋊S3), D4.10(C2×C3⋊S3), Q8.16(C2×C3⋊S3), C4.34(C22×C3⋊S3), C2.14(C23×C3⋊S3), (C32×C4○D4)⋊10C2, (C4×C3⋊S3).49C22, (C2×C3⋊S3).57C23, (C2×C6).19(C22×S3), (C2×C32⋊4Q8)⋊23C2, C22.4(C22×C3⋊S3), (C2×C3⋊Dic3).106C22, (C2×C4).24(C2×C3⋊S3), SmallGroup(288,1015)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1348 in 438 conjugacy classes, 153 normal (12 characteristic)
C1, C2, C2 [×5], C3 [×4], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×8], C6 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C32, Dic3 [×24], C12 [×16], D6 [×8], C2×C6 [×12], C2×Q8 [×5], C4○D4, C4○D4 [×9], C3⋊S3 [×2], C3×C6, C3×C6 [×3], Dic6 [×36], C4×S3 [×24], D12 [×4], C2×Dic3 [×24], C3⋊D4 [×24], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], 2- (1+4), C3⋊Dic3 [×6], C3×C12, C3×C12 [×3], C2×C3⋊S3 [×2], C62 [×3], C2×Dic6 [×12], C4○D12 [×12], D4⋊2S3 [×24], S3×Q8 [×8], C3×C4○D4 [×4], C32⋊4Q8 [×9], C4×C3⋊S3 [×6], C12⋊S3, C2×C3⋊Dic3 [×6], C32⋊7D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, Q8○D12 [×4], C2×C32⋊4Q8 [×3], C12.59D6 [×3], C12.D6 [×6], Q8×C3⋊S3 [×2], C32×C4○D4, C32⋊92- (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○D12 [×4], C23×C3⋊S3, C32⋊92- (1+4)
Generators and relations
G = < a,b,c,d,e,f | a3=b3=c4=d2=1, e2=f2=c2, ab=ba, cac-1=eae-1=a-1, ad=da, af=fa, cbc-1=ebe-1=b-1, bd=db, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >
►On 144 points
Generators in S144
(1 73 94)(2 95 74)(3 75 96)(4 93 76)(5 71 136)(6 133 72)(7 69 134)(8 135 70)(9 14 40)(10 37 15)(11 16 38)(12 39 13)(17 98 41)(18 42 99)(19 100 43)(20 44 97)(21 81 65)(22 66 82)(23 83 67)(24 68 84)(25 121 130)(26 131 122)(27 123 132)(28 129 124)(29 119 59)(30 60 120)(31 117 57)(32 58 118)(33 54 125)(34 126 55)(35 56 127)(36 128 53)(45 64 52)(46 49 61)(47 62 50)(48 51 63)(77 113 85)(78 86 114)(79 115 87)(80 88 116)(89 144 102)(90 103 141)(91 142 104)(92 101 143)(105 111 137)(106 138 112)(107 109 139)(108 140 110)
(1 123 34)(2 35 124)(3 121 36)(4 33 122)(5 48 109)(6 110 45)(7 46 111)(8 112 47)(9 42 103)(10 104 43)(11 44 101)(12 102 41)(13 144 98)(14 99 141)(15 142 100)(16 97 143)(17 39 89)(18 90 40)(19 37 91)(20 92 38)(21 115 60)(22 57 116)(23 113 58)(24 59 114)(25 53 96)(26 93 54)(27 55 94)(28 95 56)(29 78 68)(30 65 79)(31 80 66)(32 67 77)(49 137 69)(50 70 138)(51 139 71)(52 72 140)(61 105 134)(62 135 106)(63 107 136)(64 133 108)(73 132 126)(74 127 129)(75 130 128)(76 125 131)(81 87 120)(82 117 88)(83 85 118)(84 119 86)
(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)(5 140)(6 139)(7 138)(8 137)(9 12)(10 11)(13 40)(14 39)(15 38)(16 37)(17 141)(18 144)(19 143)(20 142)(21 86)(22 85)(23 88)(24 87)(25 127)(26 126)(27 125)(28 128)(29 30)(31 32)(33 123)(34 122)(35 121)(36 124)(41 103)(42 102)(43 101)(44 104)(45 51)(46 50)(47 49)(48 52)(53 129)(54 132)(55 131)(56 130)(57 118)(58 117)(59 120)(60 119)(61 62)(63 64)(65 78)(66 77)(67 80)(68 79)(69 112)(70 111)(71 110)(72 109)(73 93)(74 96)(75 95)(76 94)(81 114)(82 113)(83 116)(84 115)(89 99)(90 98)(91 97)(92 100)(105 135)(106 134)(107 133)(108 136)
(1 11 3 9)(2 12 4 10)(5 114 7 116)(6 115 8 113)(13 93 15 95)(14 94 16 96)(17 125 19 127)(18 126 20 128)(21 112 23 110)(22 109 24 111)(25 141 27 143)(26 142 28 144)(29 61 31 63)(30 62 32 64)(33 43 35 41)(34 44 36 42)(37 74 39 76)(38 75 40 73)(45 60 47 58)(46 57 48 59)(49 117 51 119)(50 118 52 120)(53 99 55 97)(54 100 56 98)(65 106 67 108)(66 107 68 105)(69 88 71 86)(70 85 72 87)(77 133 79 135)(78 134 80 136)(81 138 83 140)(82 139 84 137)(89 131 91 129)(90 132 92 130)(101 121 103 123)(102 122 104 124)
(1 32 3 30)(2 29 4 31)(5 19 7 17)(6 20 8 18)(9 64 11 62)(10 61 12 63)(13 51 15 49)(14 52 16 50)(21 132 23 130)(22 129 24 131)(25 81 27 83)(26 82 28 84)(33 80 35 78)(34 77 36 79)(37 46 39 48)(38 47 40 45)(41 136 43 134)(42 133 44 135)(53 87 55 85)(54 88 56 86)(57 74 59 76)(58 75 60 73)(65 123 67 121)(66 124 68 122)(69 98 71 100)(70 99 72 97)(89 109 91 111)(90 110 92 112)(93 117 95 119)(94 118 96 120)(101 106 103 108)(102 107 104 105)(113 128 115 126)(114 125 116 127)(137 144 139 142)(138 141 140 143)
G:=sub<Sym(144)| (1,73,94)(2,95,74)(3,75,96)(4,93,76)(5,71,136)(6,133,72)(7,69,134)(8,135,70)(9,14,40)(10,37,15)(11,16,38)(12,39,13)(17,98,41)(18,42,99)(19,100,43)(20,44,97)(21,81,65)(22,66,82)(23,83,67)(24,68,84)(25,121,130)(26,131,122)(27,123,132)(28,129,124)(29,119,59)(30,60,120)(31,117,57)(32,58,118)(33,54,125)(34,126,55)(35,56,127)(36,128,53)(45,64,52)(46,49,61)(47,62,50)(48,51,63)(77,113,85)(78,86,114)(79,115,87)(80,88,116)(89,144,102)(90,103,141)(91,142,104)(92,101,143)(105,111,137)(106,138,112)(107,109,139)(108,140,110), (1,123,34)(2,35,124)(3,121,36)(4,33,122)(5,48,109)(6,110,45)(7,46,111)(8,112,47)(9,42,103)(10,104,43)(11,44,101)(12,102,41)(13,144,98)(14,99,141)(15,142,100)(16,97,143)(17,39,89)(18,90,40)(19,37,91)(20,92,38)(21,115,60)(22,57,116)(23,113,58)(24,59,114)(25,53,96)(26,93,54)(27,55,94)(28,95,56)(29,78,68)(30,65,79)(31,80,66)(32,67,77)(49,137,69)(50,70,138)(51,139,71)(52,72,140)(61,105,134)(62,135,106)(63,107,136)(64,133,108)(73,132,126)(74,127,129)(75,130,128)(76,125,131)(81,87,120)(82,117,88)(83,85,118)(84,119,86), (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)(5,140)(6,139)(7,138)(8,137)(9,12)(10,11)(13,40)(14,39)(15,38)(16,37)(17,141)(18,144)(19,143)(20,142)(21,86)(22,85)(23,88)(24,87)(25,127)(26,126)(27,125)(28,128)(29,30)(31,32)(33,123)(34,122)(35,121)(36,124)(41,103)(42,102)(43,101)(44,104)(45,51)(46,50)(47,49)(48,52)(53,129)(54,132)(55,131)(56,130)(57,118)(58,117)(59,120)(60,119)(61,62)(63,64)(65,78)(66,77)(67,80)(68,79)(69,112)(70,111)(71,110)(72,109)(73,93)(74,96)(75,95)(76,94)(81,114)(82,113)(83,116)(84,115)(89,99)(90,98)(91,97)(92,100)(105,135)(106,134)(107,133)(108,136), (1,11,3,9)(2,12,4,10)(5,114,7,116)(6,115,8,113)(13,93,15,95)(14,94,16,96)(17,125,19,127)(18,126,20,128)(21,112,23,110)(22,109,24,111)(25,141,27,143)(26,142,28,144)(29,61,31,63)(30,62,32,64)(33,43,35,41)(34,44,36,42)(37,74,39,76)(38,75,40,73)(45,60,47,58)(46,57,48,59)(49,117,51,119)(50,118,52,120)(53,99,55,97)(54,100,56,98)(65,106,67,108)(66,107,68,105)(69,88,71,86)(70,85,72,87)(77,133,79,135)(78,134,80,136)(81,138,83,140)(82,139,84,137)(89,131,91,129)(90,132,92,130)(101,121,103,123)(102,122,104,124), (1,32,3,30)(2,29,4,31)(5,19,7,17)(6,20,8,18)(9,64,11,62)(10,61,12,63)(13,51,15,49)(14,52,16,50)(21,132,23,130)(22,129,24,131)(25,81,27,83)(26,82,28,84)(33,80,35,78)(34,77,36,79)(37,46,39,48)(38,47,40,45)(41,136,43,134)(42,133,44,135)(53,87,55,85)(54,88,56,86)(57,74,59,76)(58,75,60,73)(65,123,67,121)(66,124,68,122)(69,98,71,100)(70,99,72,97)(89,109,91,111)(90,110,92,112)(93,117,95,119)(94,118,96,120)(101,106,103,108)(102,107,104,105)(113,128,115,126)(114,125,116,127)(137,144,139,142)(138,141,140,143)>;
G:=Group( (1,73,94)(2,95,74)(3,75,96)(4,93,76)(5,71,136)(6,133,72)(7,69,134)(8,135,70)(9,14,40)(10,37,15)(11,16,38)(12,39,13)(17,98,41)(18,42,99)(19,100,43)(20,44,97)(21,81,65)(22,66,82)(23,83,67)(24,68,84)(25,121,130)(26,131,122)(27,123,132)(28,129,124)(29,119,59)(30,60,120)(31,117,57)(32,58,118)(33,54,125)(34,126,55)(35,56,127)(36,128,53)(45,64,52)(46,49,61)(47,62,50)(48,51,63)(77,113,85)(78,86,114)(79,115,87)(80,88,116)(89,144,102)(90,103,141)(91,142,104)(92,101,143)(105,111,137)(106,138,112)(107,109,139)(108,140,110), (1,123,34)(2,35,124)(3,121,36)(4,33,122)(5,48,109)(6,110,45)(7,46,111)(8,112,47)(9,42,103)(10,104,43)(11,44,101)(12,102,41)(13,144,98)(14,99,141)(15,142,100)(16,97,143)(17,39,89)(18,90,40)(19,37,91)(20,92,38)(21,115,60)(22,57,116)(23,113,58)(24,59,114)(25,53,96)(26,93,54)(27,55,94)(28,95,56)(29,78,68)(30,65,79)(31,80,66)(32,67,77)(49,137,69)(50,70,138)(51,139,71)(52,72,140)(61,105,134)(62,135,106)(63,107,136)(64,133,108)(73,132,126)(74,127,129)(75,130,128)(76,125,131)(81,87,120)(82,117,88)(83,85,118)(84,119,86), (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)(5,140)(6,139)(7,138)(8,137)(9,12)(10,11)(13,40)(14,39)(15,38)(16,37)(17,141)(18,144)(19,143)(20,142)(21,86)(22,85)(23,88)(24,87)(25,127)(26,126)(27,125)(28,128)(29,30)(31,32)(33,123)(34,122)(35,121)(36,124)(41,103)(42,102)(43,101)(44,104)(45,51)(46,50)(47,49)(48,52)(53,129)(54,132)(55,131)(56,130)(57,118)(58,117)(59,120)(60,119)(61,62)(63,64)(65,78)(66,77)(67,80)(68,79)(69,112)(70,111)(71,110)(72,109)(73,93)(74,96)(75,95)(76,94)(81,114)(82,113)(83,116)(84,115)(89,99)(90,98)(91,97)(92,100)(105,135)(106,134)(107,133)(108,136), (1,11,3,9)(2,12,4,10)(5,114,7,116)(6,115,8,113)(13,93,15,95)(14,94,16,96)(17,125,19,127)(18,126,20,128)(21,112,23,110)(22,109,24,111)(25,141,27,143)(26,142,28,144)(29,61,31,63)(30,62,32,64)(33,43,35,41)(34,44,36,42)(37,74,39,76)(38,75,40,73)(45,60,47,58)(46,57,48,59)(49,117,51,119)(50,118,52,120)(53,99,55,97)(54,100,56,98)(65,106,67,108)(66,107,68,105)(69,88,71,86)(70,85,72,87)(77,133,79,135)(78,134,80,136)(81,138,83,140)(82,139,84,137)(89,131,91,129)(90,132,92,130)(101,121,103,123)(102,122,104,124), (1,32,3,30)(2,29,4,31)(5,19,7,17)(6,20,8,18)(9,64,11,62)(10,61,12,63)(13,51,15,49)(14,52,16,50)(21,132,23,130)(22,129,24,131)(25,81,27,83)(26,82,28,84)(33,80,35,78)(34,77,36,79)(37,46,39,48)(38,47,40,45)(41,136,43,134)(42,133,44,135)(53,87,55,85)(54,88,56,86)(57,74,59,76)(58,75,60,73)(65,123,67,121)(66,124,68,122)(69,98,71,100)(70,99,72,97)(89,109,91,111)(90,110,92,112)(93,117,95,119)(94,118,96,120)(101,106,103,108)(102,107,104,105)(113,128,115,126)(114,125,116,127)(137,144,139,142)(138,141,140,143) );
G=PermutationGroup([(1,73,94),(2,95,74),(3,75,96),(4,93,76),(5,71,136),(6,133,72),(7,69,134),(8,135,70),(9,14,40),(10,37,15),(11,16,38),(12,39,13),(17,98,41),(18,42,99),(19,100,43),(20,44,97),(21,81,65),(22,66,82),(23,83,67),(24,68,84),(25,121,130),(26,131,122),(27,123,132),(28,129,124),(29,119,59),(30,60,120),(31,117,57),(32,58,118),(33,54,125),(34,126,55),(35,56,127),(36,128,53),(45,64,52),(46,49,61),(47,62,50),(48,51,63),(77,113,85),(78,86,114),(79,115,87),(80,88,116),(89,144,102),(90,103,141),(91,142,104),(92,101,143),(105,111,137),(106,138,112),(107,109,139),(108,140,110)], [(1,123,34),(2,35,124),(3,121,36),(4,33,122),(5,48,109),(6,110,45),(7,46,111),(8,112,47),(9,42,103),(10,104,43),(11,44,101),(12,102,41),(13,144,98),(14,99,141),(15,142,100),(16,97,143),(17,39,89),(18,90,40),(19,37,91),(20,92,38),(21,115,60),(22,57,116),(23,113,58),(24,59,114),(25,53,96),(26,93,54),(27,55,94),(28,95,56),(29,78,68),(30,65,79),(31,80,66),(32,67,77),(49,137,69),(50,70,138),(51,139,71),(52,72,140),(61,105,134),(62,135,106),(63,107,136),(64,133,108),(73,132,126),(74,127,129),(75,130,128),(76,125,131),(81,87,120),(82,117,88),(83,85,118),(84,119,86)], [(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),(5,140),(6,139),(7,138),(8,137),(9,12),(10,11),(13,40),(14,39),(15,38),(16,37),(17,141),(18,144),(19,143),(20,142),(21,86),(22,85),(23,88),(24,87),(25,127),(26,126),(27,125),(28,128),(29,30),(31,32),(33,123),(34,122),(35,121),(36,124),(41,103),(42,102),(43,101),(44,104),(45,51),(46,50),(47,49),(48,52),(53,129),(54,132),(55,131),(56,130),(57,118),(58,117),(59,120),(60,119),(61,62),(63,64),(65,78),(66,77),(67,80),(68,79),(69,112),(70,111),(71,110),(72,109),(73,93),(74,96),(75,95),(76,94),(81,114),(82,113),(83,116),(84,115),(89,99),(90,98),(91,97),(92,100),(105,135),(106,134),(107,133),(108,136)], [(1,11,3,9),(2,12,4,10),(5,114,7,116),(6,115,8,113),(13,93,15,95),(14,94,16,96),(17,125,19,127),(18,126,20,128),(21,112,23,110),(22,109,24,111),(25,141,27,143),(26,142,28,144),(29,61,31,63),(30,62,32,64),(33,43,35,41),(34,44,36,42),(37,74,39,76),(38,75,40,73),(45,60,47,58),(46,57,48,59),(49,117,51,119),(50,118,52,120),(53,99,55,97),(54,100,56,98),(65,106,67,108),(66,107,68,105),(69,88,71,86),(70,85,72,87),(77,133,79,135),(78,134,80,136),(81,138,83,140),(82,139,84,137),(89,131,91,129),(90,132,92,130),(101,121,103,123),(102,122,104,124)], [(1,32,3,30),(2,29,4,31),(5,19,7,17),(6,20,8,18),(9,64,11,62),(10,61,12,63),(13,51,15,49),(14,52,16,50),(21,132,23,130),(22,129,24,131),(25,81,27,83),(26,82,28,84),(33,80,35,78),(34,77,36,79),(37,46,39,48),(38,47,40,45),(41,136,43,134),(42,133,44,135),(53,87,55,85),(54,88,56,86),(57,74,59,76),(58,75,60,73),(65,123,67,121),(66,124,68,122),(69,98,71,100),(70,99,72,97),(89,109,91,111),(90,110,92,112),(93,117,95,119),(94,118,96,120),(101,106,103,108),(102,107,104,105),(113,128,115,126),(114,125,116,127),(137,144,139,142),(138,141,140,143)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 3 | 0 | 0 | 0 | 0 |
| 12 | 11 | 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 | 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 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 11 | 11 |
| 0 | 0 | 7 | 3 | 9 | 2 |
| 0 | 0 | 9 | 9 | 3 | 3 |
| 0 | 0 | 5 | 4 | 6 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 11 | 4 |
| 0 | 0 | 7 | 3 | 9 | 2 |
| 0 | 0 | 9 | 8 | 3 | 7 |
| 0 | 0 | 5 | 4 | 6 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 11 | 2 |
| 0 | 0 | 0 | 12 | 0 | 2 |
| 0 | 0 | 1 | 12 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 8 | 0 |
| 0 | 0 | 0 | 12 | 0 | 8 |
| 0 | 0 | 3 | 0 | 1 | 0 |
| 0 | 0 | 0 | 3 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,3,11,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,10,7,9,5,0,0,10,3,9,4,0,0,11,9,3,6,0,0,11,2,3,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,9,5,0,0,6,3,8,4,0,0,11,9,3,6,0,0,4,2,7,10],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,12,12,12,12,0,0,11,0,12,0,0,0,2,2,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,3,0,0,0,0,12,0,3,0,0,8,0,1,0,0,0,0,8,0,1] >;
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2- (1+4) | Q8○D12 |
| kernel | C32⋊92- (1+4) | C2×C32⋊4Q8 | C12.59D6 | C12.D6 | Q8×C3⋊S3 | C32×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C32 | C3 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 12 | 12 | 4 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_3^2\rtimes_92_-^{(1+4)} % in TeX
G:=Group("C3^2:9ES-(2,2)"); // GroupNames label
G:=SmallGroup(288,1015);
// by ID
G=gap.SmallGroup(288,1015);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,2693,9414]);
// 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,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,a*f=f*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,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
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