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G = C23.233C24  order 128 = 27

86th central extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.233C24, C22.502- 1+4, Q8⋊6(C4⋊C4), (C4×Q8)⋊20C4, (C2×Q8).27Q8, (C2×Q8).261D4, C42.187(C2×C4), C2.3(Q8⋊5D4), C2.2(Q8⋊3Q8), C42⋊8C4.22C2, Q8○3(C2.C42), C22.41(C22×Q8), (C2×C42).432C22, C22.124(C23×C4), C22.108(C22×D4), (C22×C4).1248C23, (C22×Q8).510C22, C2.C42.59C22, C23.65C23.34C2, C2.15(C23.32C23), C4.19(C2×C4⋊C4), (C4×C4⋊C4).39C2, (C2×C4×Q8).24C2, C2.29(C4×C4○D4), C4⋊C4.209(C2×C4), C2.17(C22×C4⋊C4), (C2×C4).302(C2×Q8), (C2×C4).1072(C2×D4), (C2×Q8).220(C2×C4), (C2×C4).852(C4○D4), (C2×C4⋊C4).822C22, (C2×C4).492(C22×C4), C22.118(C2×C4○D4), (C2×Q8)○2(C2.C42), SmallGroup(128,1083)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23.233C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×Q8 — C23.233C24
Lower central
C1 — C22 — C23.233C24
Upper central
C1 — C23 — C23.233C24
Jennings
C1 — C23 — C23.233C24

Generators and relations for C23.233C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=g2=a, f2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 380 in 276 conjugacy classes, 184 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C4×C4⋊C4, C42⋊8C4, C23.65C23, C2×C4×Q8, C2×C4×Q8, C23.233C24
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C22×C4⋊C4, C4×C4○D4, C23.32C23, Q8⋊5D4, Q8⋊3Q8, C23.233C24

Smallest permutation representation of C23.233C24
►Regular action on 128 points
Generators in S128
(1 11)(2 12)(3 9)(4 10)(5 70)(6 71)(7 72)(8 69)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 127)(66 128)(67 125)(68 126)

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

(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 39)(38 40)(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 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)

(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)

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

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

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

50 conjugacy classes

class 1 2A···2G4A···4X4Y···4AP
order12···24···44···4
size11···12···24···4

50 irreducible representations

dim1111112224
type++++++--
imageC1C2C2C2C2C4D4Q8C4○D42- 1+4
kernelC23.233C24C4×C4⋊C4C42⋊8C4C23.65C23C2×C4×Q8C4×Q8C2×Q8C2×Q8C2×C4C22
# reps13363164482

Matrix representation of C23.233C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
020000
300000
003000
003200
000042
000041
,
200000
030000
001300
001400
000040
000004
,
200000
020000
003000
000300
000030
000032
,
010000
400000
003000
003200
000010
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,3,0,0,0,0,0,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C23.233C24 in GAP, Magma, Sage, TeX 

C_2^3._{233}C_2^4
 
% in TeX
 
G:=Group("C2^3.233C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1083);
 
// by ID
 
G=gap.SmallGroup(128,1083);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,268,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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