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G = C32×C22⋊Q8order 288 = 25·32

Direct product of C32 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C32×C22⋊Q8, C628Q8, C23.12C62, C62.291C23, (C6×Q8)⋊12C6, C6.86(C6×D4), C6.26(C6×Q8), C12.79(C3×D4), (C2×C4).4C62, (C3×C12).180D4, C4.13(D4×C32), C222(Q8×C32), (C22×C12).36C6, (C6×C12).371C22, C22.12(C2×C62), (C2×C62).89C22, C4⋊C43(C3×C6), C2.6(D4×C3×C6), C2.3(Q8×C3×C6), (C3×C4⋊C4)⋊12C6, (Q8×C3×C6)⋊15C2, (C2×C6)⋊4(C3×Q8), (C2×C6×C12).25C2, (C2×Q8)⋊3(C3×C6), C6.52(C3×C4○D4), (C3×C6).80(C2×Q8), (C32×C4⋊C4)⋊21C2, (C2×C12).97(C2×C6), (C3×C6).303(C2×D4), C22⋊C4.1(C3×C6), (C22×C4).7(C3×C6), C2.5(C32×C4○D4), (C3×C22⋊C4).11C6, (C2×C6).97(C22×C6), (C22×C6).53(C2×C6), (C3×C6).169(C4○D4), (C32×C22⋊C4).7C2, SmallGroup(288,819)

Series: Derived Chief Lower central Upper central

C1C22 — C32×C22⋊Q8
C1C2C22C2×C6C62C6×C12Q8×C3×C6 — C32×C22⋊Q8
C1C22 — C32×C22⋊Q8
C1C62 — C32×C22⋊Q8

Generators and relations for C32×C22⋊Q8
 G = < a,b,c,d,e,f | a3=b3=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 300 in 222 conjugacy classes, 144 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], Q8 [×2], C23, C32, C12 [×8], C12 [×20], C2×C6 [×12], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], C2×C12 [×24], C2×C12 [×8], C3×Q8 [×8], C22×C6 [×4], C22⋊Q8, C3×C12 [×2], C3×C12 [×5], C62, C62 [×2], C62 [×2], C3×C22⋊C4 [×8], C3×C4⋊C4 [×12], C22×C12 [×4], C6×Q8 [×4], C6×C12 [×2], C6×C12 [×4], C6×C12 [×2], Q8×C32 [×2], C2×C62, C3×C22⋊Q8 [×4], C32×C22⋊C4 [×2], C32×C4⋊C4, C32×C4⋊C4 [×2], C2×C6×C12, Q8×C3×C6, C32×C22⋊Q8
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], Q8 [×2], C23, C32, C2×C6 [×28], C2×D4, C2×Q8, C4○D4, C3×C6 [×7], C3×D4 [×8], C3×Q8 [×8], C22×C6 [×4], C22⋊Q8, C62 [×7], C6×D4 [×4], C6×Q8 [×4], C3×C4○D4 [×4], D4×C32 [×2], Q8×C32 [×2], C2×C62, C3×C22⋊Q8 [×4], D4×C3×C6, Q8×C3×C6, C32×C4○D4, C32×C22⋊Q8

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

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

G:=Group( (1,5,15)(2,6,16)(3,7,13)(4,8,14)(9,53,45)(10,54,46)(11,55,47)(12,56,48)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,75,29)(22,76,30)(23,73,31)(24,74,32)(25,69,61)(26,70,62)(27,71,63)(28,72,64)(33,65,57)(34,66,58)(35,67,59)(36,68,60)(37,104,96)(38,101,93)(39,102,94)(40,103,95)(77,112,85)(78,109,86)(79,110,87)(80,111,88)(81,127,119)(82,128,120)(83,125,117)(84,126,118)(89,123,115)(90,124,116)(91,121,113)(92,122,114)(97,143,135)(98,144,136)(99,141,133)(100,142,134)(105,139,131)(106,140,132)(107,137,129)(108,138,130), (1,11,43)(2,12,44)(3,9,41)(4,10,42)(5,55,19)(6,56,20)(7,53,17)(8,54,18)(13,45,49)(14,46,50)(15,47,51)(16,48,52)(21,25,57)(22,26,58)(23,27,59)(24,28,60)(29,61,65)(30,62,66)(31,63,67)(32,64,68)(33,75,69)(34,76,70)(35,73,71)(36,74,72)(37,142,108)(38,143,105)(39,144,106)(40,141,107)(77,81,115)(78,82,116)(79,83,113)(80,84,114)(85,119,123)(86,120,124)(87,117,121)(88,118,122)(89,112,127)(90,109,128)(91,110,125)(92,111,126)(93,97,131)(94,98,132)(95,99,129)(96,100,130)(101,135,139)(102,136,140)(103,133,137)(104,134,138), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,111)(38,112)(39,109)(40,110)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)(128,144), (1,23)(2,24)(3,21)(4,22)(5,73)(6,74)(7,75)(8,76)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(37,109)(38,110)(39,111)(40,112)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(77,95)(78,96)(79,93)(80,94)(81,99)(82,100)(83,97)(84,98)(85,103)(86,104)(87,101)(88,102)(89,107)(90,108)(91,105)(92,106)(113,131)(114,132)(115,129)(116,130)(117,135)(118,136)(119,133)(120,134)(121,139)(122,140)(123,137)(124,138)(125,143)(126,144)(127,141)(128,142), (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), (1,79,3,77)(2,78,4,80)(5,110,7,112)(6,109,8,111)(9,81,11,83)(10,84,12,82)(13,85,15,87)(14,88,16,86)(17,89,19,91)(18,92,20,90)(21,95,23,93)(22,94,24,96)(25,99,27,97)(26,98,28,100)(29,103,31,101)(30,102,32,104)(33,107,35,105)(34,106,36,108)(37,76,39,74)(38,75,40,73)(41,115,43,113)(42,114,44,116)(45,119,47,117)(46,118,48,120)(49,123,51,121)(50,122,52,124)(53,127,55,125)(54,126,56,128)(57,129,59,131)(58,132,60,130)(61,133,63,135)(62,136,64,134)(65,137,67,139)(66,140,68,138)(69,141,71,143)(70,144,72,142) );

G=PermutationGroup([(1,5,15),(2,6,16),(3,7,13),(4,8,14),(9,53,45),(10,54,46),(11,55,47),(12,56,48),(17,49,41),(18,50,42),(19,51,43),(20,52,44),(21,75,29),(22,76,30),(23,73,31),(24,74,32),(25,69,61),(26,70,62),(27,71,63),(28,72,64),(33,65,57),(34,66,58),(35,67,59),(36,68,60),(37,104,96),(38,101,93),(39,102,94),(40,103,95),(77,112,85),(78,109,86),(79,110,87),(80,111,88),(81,127,119),(82,128,120),(83,125,117),(84,126,118),(89,123,115),(90,124,116),(91,121,113),(92,122,114),(97,143,135),(98,144,136),(99,141,133),(100,142,134),(105,139,131),(106,140,132),(107,137,129),(108,138,130)], [(1,11,43),(2,12,44),(3,9,41),(4,10,42),(5,55,19),(6,56,20),(7,53,17),(8,54,18),(13,45,49),(14,46,50),(15,47,51),(16,48,52),(21,25,57),(22,26,58),(23,27,59),(24,28,60),(29,61,65),(30,62,66),(31,63,67),(32,64,68),(33,75,69),(34,76,70),(35,73,71),(36,74,72),(37,142,108),(38,143,105),(39,144,106),(40,141,107),(77,81,115),(78,82,116),(79,83,113),(80,84,114),(85,119,123),(86,120,124),(87,117,121),(88,118,122),(89,112,127),(90,109,128),(91,110,125),(92,111,126),(93,97,131),(94,98,132),(95,99,129),(96,100,130),(101,135,139),(102,136,140),(103,133,137),(104,134,138)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,111),(38,112),(39,109),(40,110),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,93),(78,94),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(121,137),(122,138),(123,139),(124,140),(125,141),(126,142),(127,143),(128,144)], [(1,23),(2,24),(3,21),(4,22),(5,73),(6,74),(7,75),(8,76),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(37,109),(38,110),(39,111),(40,112),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(77,95),(78,96),(79,93),(80,94),(81,99),(82,100),(83,97),(84,98),(85,103),(86,104),(87,101),(88,102),(89,107),(90,108),(91,105),(92,106),(113,131),(114,132),(115,129),(116,130),(117,135),(118,136),(119,133),(120,134),(121,139),(122,140),(123,137),(124,138),(125,143),(126,144),(127,141),(128,142)], [(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)], [(1,79,3,77),(2,78,4,80),(5,110,7,112),(6,109,8,111),(9,81,11,83),(10,84,12,82),(13,85,15,87),(14,88,16,86),(17,89,19,91),(18,92,20,90),(21,95,23,93),(22,94,24,96),(25,99,27,97),(26,98,28,100),(29,103,31,101),(30,102,32,104),(33,107,35,105),(34,106,36,108),(37,76,39,74),(38,75,40,73),(41,115,43,113),(42,114,44,116),(45,119,47,117),(46,118,48,120),(49,123,51,121),(50,122,52,124),(53,127,55,125),(54,126,56,128),(57,129,59,131),(58,132,60,130),(61,133,63,135),(62,136,64,134),(65,137,67,139),(66,140,68,138),(69,141,71,143),(70,144,72,142)])

126 conjugacy classes

class 1 2A2B2C2D2E3A···3H4A4B4C4D4E4F4G4H6A···6X6Y···6AN12A···12AF12AG···12BL
order1222223···3444444446···66···612···1212···12
size1111221···1222244441···12···22···24···4

126 irreducible representations

dim1111111111222222
type++++++-
imageC1C2C2C2C2C3C6C6C6C6D4Q8C4○D4C3×D4C3×Q8C3×C4○D4
kernelC32×C22⋊Q8C32×C22⋊C4C32×C4⋊C4C2×C6×C12Q8×C3×C6C3×C22⋊Q8C3×C22⋊C4C3×C4⋊C4C22×C12C6×Q8C3×C12C62C3×C6C12C2×C6C6
# reps123118162488222161616

Matrix representation of C32×C22⋊Q8 in GL4(𝔽13) generated by

9000
0900
0010
0001
,
1000
0100
0030
0003
,
12000
01200
0010
001212
,
1000
0100
00120
00012
,
11100
11200
0010
0001
,
1600
41200
001211
0001
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,1,0,0,11,12,0,0,0,0,1,0,0,0,0,1],[1,4,0,0,6,12,0,0,0,0,12,0,0,0,11,1] >;

C32×C22⋊Q8 in GAP, Magma, Sage, TeX

C_3^2\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C3^2xC2^2:Q8");
// GroupNames label

G:=SmallGroup(288,819);
// by ID

G=gap.SmallGroup(288,819);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1037,512,3110]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=e^4=1,f^2=e^2,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,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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