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G = C23.626C24  order 128 = 27

343rd central stem extension by C23 of C24

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

Aliases: C23.626C24, C22.3012- 1+4, C22.3992+ 1+4, C4⋊C4.8Q8, C2.32(D4×Q8), C4⋊C4.128D4, C2.77(D4⋊6D4), C2.27(Q8⋊3Q8), (C22×C4).888C23, (C2×C42).677C22, C22.435(C22×D4), C22.148(C22×Q8), C2.C42.332C22, C23.63C23.42C2, C23.81C23.33C2, C23.65C23.73C2, C2.3(C22.58C24), C2.33(C23.41C23), C2.77(C22.33C24), C2.54(C22.31C24), (C2×C4).72(C2×Q8), (C2×C4).121(C2×D4), (C2×C4).207(C4○D4), (C2×C4⋊C4).439C22, C22.488(C2×C4○D4), (C2×C42.C2).28C2, SmallGroup(128,1458)

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

Derived series
C1 — C23 — C23.626C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C23.626C24
Lower central
C1 — C23 — C23.626C24
Upper central
C1 — C23 — C23.626C24
Jennings
C1 — C23 — C23.626C24

Generators and relations for C23.626C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=ba=ab, f2=b, g2=a, 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, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 340 in 208 conjugacy classes, 104 normal (30 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C23.63C23, C23.65C23, C23.65C23, C23.81C23, C2×C42.C2, C23.626C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.31C24, C22.33C24, C23.41C23, D4⋊6D4, D4×Q8, Q8⋊3Q8, C22.58C24, C23.626C24

Smallest permutation representation of C23.626C24
►Regular action on 128 points
Generators in S128
(1 73)(2 74)(3 75)(4 76)(5 126)(6 127)(7 128)(8 125)(9 39)(10 40)(11 37)(12 38)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(41 47)(42 48)(43 45)(44 46)(49 55)(50 56)(51 53)(52 54)(57 63)(58 64)(59 61)(60 62)(65 71)(66 72)(67 69)(68 70)(101 107)(102 108)(103 105)(104 106)(109 115)(110 116)(111 113)(112 114)(117 123)(118 124)(119 121)(120 122)

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

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

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

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

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

32 conjugacy classes

class 1 2A···2G4A···4R4S···4X
order12···24···44···4
size11···14···48···8

32 irreducible representations

dim1111122244
type++++++-+-
imageC1C2C2C2C2D4Q8C4○D42+ 1+42- 1+4
kernelC23.626C24C23.63C23C23.65C23C23.81C23C2×C42.C2C4⋊C4C4⋊C4C2×C4C22C22
# reps1256244413

Matrix representation of C23.626C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
110000
340000
000100
004000
000010
000001
,
220000
030000
003000
000300
000032
000012
,
100000
010000
003000
000200
000010
000024
,
300000
420000
001000
000400
000040
000004

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

C23.626C24 in GAP, Magma, Sage, TeX 

C_2^3._{626}C_2^4
 
% in TeX
 
G:=Group("C2^3.626C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1458);
 
// by ID
 
G=gap.SmallGroup(128,1458);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=b*a=a*b,f^2=b,g^2=a,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,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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