metabelian, supersoluble, monomial
Aliases: C62.225C23, C32⋊19(C4×D4), C3⋊Dic3⋊17D4, C62⋊13(C2×C4), C6.107(S3×D4), C32⋊7D4⋊3C4, (C2×C12).206D6, (C22×C6).87D6, C3⋊4(Dic3⋊4D4), C6.94(D4⋊2S3), C6.11D12⋊20C2, (C6×C12).251C22, C6.Dic6⋊20C2, (C2×C62).64C22, C2.2(C12.D6), C6.67(S3×C2×C4), C2.2(D4×C3⋊S3), (C2×C6)⋊10(C4×S3), C22⋊2(C4×C3⋊S3), (C3×C22⋊C4)⋊9S3, C3⋊Dic3⋊8(C2×C4), C22⋊C4⋊7(C3⋊S3), (C4×C3⋊Dic3)⋊22C2, (C3×C6).230(C2×D4), C23.19(C2×C3⋊S3), (C3×C6).98(C22×C4), (C22×C3⋊Dic3)⋊5C2, (C2×C32⋊7D4).9C2, (C3×C6).143(C4○D4), (C32×C22⋊C4)⋊17C2, (C2×C6).242(C22×S3), C22.14(C22×C3⋊S3), (C22×C3⋊S3).82C22, (C2×C3⋊Dic3).155C22, C2.9(C2×C4×C3⋊S3), (C2×C4×C3⋊S3)⋊18C2, (C2×C3⋊S3)⋊10(C2×C4), (C2×C4).27(C2×C3⋊S3), SmallGroup(288,738)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C62.225C23 |
Generators and relations for C62.225C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ce=ec, ede-1=b3d >
Subgroups: 996 in 282 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C22×C3⋊S3, C2×C62, Dic3⋊4D4, C4×C3⋊Dic3, C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C32⋊7D4, C62.225C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3, C22×S3, C4×D4, C2×C3⋊S3, S3×C2×C4, S3×D4, D4⋊2S3, C4×C3⋊S3, C22×C3⋊S3, Dic3⋊4D4, C2×C4×C3⋊S3, D4×C3⋊S3, C12.D6, C62.225C23
(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 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)
(2 6)(3 5)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 17)(14 16)(25 139)(26 144)(27 143)(28 142)(29 141)(30 140)(31 62)(32 61)(33 66)(34 65)(35 64)(36 63)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(44 48)(45 47)(49 51)(52 54)(67 98)(68 97)(69 102)(70 101)(71 100)(72 99)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)(103 131)(104 130)(105 129)(106 128)(107 127)(108 132)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(115 125)(116 124)(117 123)(118 122)(119 121)(120 126)
(1 82)(2 83)(3 84)(4 79)(5 80)(6 81)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 87)(14 88)(15 89)(16 90)(17 85)(18 86)(19 99)(20 100)(21 101)(22 102)(23 97)(24 98)(25 95)(26 96)(27 91)(28 92)(29 93)(30 94)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 43 4 46)(2 44 5 47)(3 45 6 48)(7 103 10 106)(8 104 11 107)(9 105 12 108)(13 54 16 51)(14 49 17 52)(15 50 18 53)(19 132 22 129)(20 127 23 130)(21 128 24 131)(25 134 28 137)(26 135 29 138)(27 136 30 133)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 76 40 73)(38 77 41 74)(39 78 42 75)(55 94 58 91)(56 95 59 92)(57 96 60 93)(61 100 64 97)(62 101 65 98)(63 102 66 99)(79 122 82 125)(80 123 83 126)(81 124 84 121)(85 120 88 117)(86 115 89 118)(87 116 90 119)(109 141 112 144)(110 142 113 139)(111 143 114 140)
G:=sub<Sym(144)| (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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (2,6)(3,5)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,17)(14,16)(25,139)(26,144)(27,143)(28,142)(29,141)(30,140)(31,62)(32,61)(33,66)(34,65)(35,64)(36,63)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(44,48)(45,47)(49,51)(52,54)(67,98)(68,97)(69,102)(70,101)(71,100)(72,99)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90)(103,131)(104,130)(105,129)(106,128)(107,127)(108,132)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(115,125)(116,124)(117,123)(118,122)(119,121)(120,126), (1,82)(2,83)(3,84)(4,79)(5,80)(6,81)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(19,99)(20,100)(21,101)(22,102)(23,97)(24,98)(25,95)(26,96)(27,91)(28,92)(29,93)(30,94)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,103,10,106)(8,104,11,107)(9,105,12,108)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,132,22,129)(20,127,23,130)(21,128,24,131)(25,134,28,137)(26,135,29,138)(27,136,30,133)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(109,141,112,144)(110,142,113,139)(111,143,114,140)>;
G:=Group( (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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (2,6)(3,5)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,17)(14,16)(25,139)(26,144)(27,143)(28,142)(29,141)(30,140)(31,62)(32,61)(33,66)(34,65)(35,64)(36,63)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(44,48)(45,47)(49,51)(52,54)(67,98)(68,97)(69,102)(70,101)(71,100)(72,99)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90)(103,131)(104,130)(105,129)(106,128)(107,127)(108,132)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(115,125)(116,124)(117,123)(118,122)(119,121)(120,126), (1,82)(2,83)(3,84)(4,79)(5,80)(6,81)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(19,99)(20,100)(21,101)(22,102)(23,97)(24,98)(25,95)(26,96)(27,91)(28,92)(29,93)(30,94)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,103,10,106)(8,104,11,107)(9,105,12,108)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,132,22,129)(20,127,23,130)(21,128,24,131)(25,134,28,137)(26,135,29,138)(27,136,30,133)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(109,141,112,144)(110,142,113,139)(111,143,114,140) );
G=PermutationGroup([[(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)], [(1,32,58,18,39,61),(2,33,59,13,40,62),(3,34,60,14,41,63),(4,35,55,15,42,64),(5,36,56,16,37,65),(6,31,57,17,38,66),(7,22,117,139,29,121),(8,23,118,140,30,122),(9,24,119,141,25,123),(10,19,120,142,26,124),(11,20,115,143,27,125),(12,21,116,144,28,126),(43,71,91,53,78,100),(44,72,92,54,73,101),(45,67,93,49,74,102),(46,68,94,50,75,97),(47,69,95,51,76,98),(48,70,96,52,77,99),(79,107,127,89,114,136),(80,108,128,90,109,137),(81,103,129,85,110,138),(82,104,130,86,111,133),(83,105,131,87,112,134),(84,106,132,88,113,135)], [(2,6),(3,5),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,17),(14,16),(25,139),(26,144),(27,143),(28,142),(29,141),(30,140),(31,62),(32,61),(33,66),(34,65),(35,64),(36,63),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(44,48),(45,47),(49,51),(52,54),(67,98),(68,97),(69,102),(70,101),(71,100),(72,99),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90),(103,131),(104,130),(105,129),(106,128),(107,127),(108,132),(109,135),(110,134),(111,133),(112,138),(113,137),(114,136),(115,125),(116,124),(117,123),(118,122),(119,121),(120,126)], [(1,82),(2,83),(3,84),(4,79),(5,80),(6,81),(7,74),(8,75),(9,76),(10,77),(11,78),(12,73),(13,87),(14,88),(15,89),(16,90),(17,85),(18,86),(19,99),(20,100),(21,101),(22,102),(23,97),(24,98),(25,95),(26,96),(27,91),(28,92),(29,93),(30,94),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(1,43,4,46),(2,44,5,47),(3,45,6,48),(7,103,10,106),(8,104,11,107),(9,105,12,108),(13,54,16,51),(14,49,17,52),(15,50,18,53),(19,132,22,129),(20,127,23,130),(21,128,24,131),(25,134,28,137),(26,135,29,138),(27,136,30,133),(31,70,34,67),(32,71,35,68),(33,72,36,69),(37,76,40,73),(38,77,41,74),(39,78,42,75),(55,94,58,91),(56,95,59,92),(57,96,60,93),(61,100,64,97),(62,101,65,98),(63,102,66,99),(79,122,82,125),(80,123,83,126),(81,124,84,121),(85,120,88,117),(86,115,89,118),(87,116,90,119),(109,141,112,144),(110,142,113,139),(111,143,114,140)]])
60 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | D4⋊2S3 |
kernel | C62.225C23 | C4×C3⋊Dic3 | C6.Dic6 | C6.11D12 | C32×C22⋊C4 | C2×C4×C3⋊S3 | C22×C3⋊Dic3 | C2×C32⋊7D4 | C32⋊7D4 | C3×C22⋊C4 | C3⋊Dic3 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 | C6 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 4 | 2 | 8 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C62.225C23 ►in GL6(𝔽13)
12 | 1 | 0 | 0 | 0 | 0 |
12 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 12 | 0 | 0 | 0 | 0 |
1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 0 | 12 |
1 | 0 | 0 | 0 | 0 | 0 |
1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 12 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 12 |
1 | 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 | 11 |
0 | 0 | 0 | 0 | 0 | 12 |
12 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 5 | 0 | 0 | 0 |
0 | 0 | 0 | 5 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;
C62.225C23 in GAP, Magma, Sage, TeX
C_6^2._{225}C_2^3
% in TeX
G:=Group("C6^2.225C2^3");
// GroupNames label
G:=SmallGroup(288,738);
// by ID
G=gap.SmallGroup(288,738);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=a^3,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=b^3*c,c*e=e*c,e*d*e^-1=b^3*d>;
// generators/relations