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G = C23.428C24  order 128 = 27

145th central stem extension by C23 of C24

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

Aliases: C23.428C24, C22.1682- 1+4, C42⋊4C4.20C2, C42⋊8C4.31C2, (C2×C42).57C22, (C22×C4).1257C23, C23.84C23.2C2, C23.63C23.25C2, C2.C42.175C22, C23.81C23.12C2, C23.83C23.12C2, C23.65C23.49C2, C2.55(C22.46C24), C2.33(C22.50C24), C2.22(C22.35C24), C2.71(C23.36C23), (C4×C4⋊C4).60C2, (C2×C4).380(C4○D4), (C2×C4⋊C4).290C22, C22.305(C2×C4○D4), SmallGroup(128,1260)

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

Derived series
C1 — C23 — C23.428C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.428C24
Lower central
C1 — C23 — C23.428C24
Upper central
C1 — C23 — C23.428C24
Jennings
C1 — C23 — C23.428C24

Generators and relations for C23.428C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=ba=ab, f2=a, g2=b, ac=ca, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 292 in 182 conjugacy classes, 92 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42⋊4C4, C4×C4⋊C4, C42⋊8C4, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C23.84C23, C23.428C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C23.36C23, C22.35C24, C22.46C24, C22.50C24, C23.428C24

Smallest permutation representation of C23.428C24
►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 55 39 87)(2 116 40 28)(3 53 37 85)(4 114 38 26)(5 82 126 50)(6 23 127 111)(7 84 128 52)(8 21 125 109)(9 113 97 25)(10 54 98 86)(11 115 99 27)(12 56 100 88)(13 117 101 29)(14 58 102 90)(15 119 103 31)(16 60 104 92)(17 121 105 33)(18 62 106 94)(19 123 107 35)(20 64 108 96)(22 68 110 70)(24 66 112 72)(30 74 118 42)(32 76 120 44)(34 78 122 46)(36 80 124 48)(41 89 73 57)(43 91 75 59)(45 93 77 61)(47 95 79 63)(49 69 81 67)(51 71 83 65)

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim11111111124
type+++++++++-
imageC1C2C2C2C2C2C2C2C2C4○D42- 1+4
kernelC23.428C24C42⋊4C4C4×C4⋊C4C42⋊8C4C23.63C23C23.65C23C23.81C23C23.83C23C23.84C23C2×C4C22
# reps111171121202

Matrix representation of C23.428C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
001000
000001
000040
,
200000
030000
004000
000100
000004
000010
,
200000
020000
003000
000200
000001
000010
,
400000
010000
002000
000300
000030
000003

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
 

C23.428C24 in GAP, Magma, Sage, TeX 

C_2^3._{428}C_2^4
 
% in TeX
 
G:=Group("C2^3.428C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1260);
 
// by ID
 
G=gap.SmallGroup(128,1260);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=b*a=a*b,f^2=a,g^2=b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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