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G = C23.490C24  order 128 = 27

207th central stem extension by C23 of C24

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

Aliases: C23.490C24, C22.2722+ 1+4, C22.2012- 1+4, C4⋊C4.25Q8, C2.25(Q8⋊3Q8), C2.48(D4⋊3Q8), (C2×C42).582C22, (C22×C4).549C23, C22.125(C22×Q8), C23.81C23.23C2, C2.C42.224C22, C23.83C23.20C2, C23.65C23.64C2, C23.63C23.33C2, C2.36(C23.37C23), C2.33(C22.33C24), C2.71(C22.46C24), C2.68(C22.47C24), C2.94(C23.36C23), (C4×C4⋊C4).75C2, (C2×C4).65(C2×Q8), (C2×C4).402(C4○D4), (C2×C4⋊C4).334C22, C22.366(C2×C4○D4), SmallGroup(128,1322)

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

Derived series
C1 — C23 — C23.490C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.490C24
Lower central
C1 — C23 — C23.490C24
Upper central
C1 — C23 — C23.490C24
Jennings
C1 — C23 — C23.490C24

Generators and relations for C23.490C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=a, e2=cb=bc, f2=c, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, 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: 308 in 190 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C4×C4⋊C4, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C23.490C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C23.37C23, C22.33C24, C22.46C24, C22.47C24, D4⋊3Q8, Q8⋊3Q8, C23.490C24

Smallest permutation representation of C23.490C24
►Regular action on 128 points
Generators in S128
(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 9)(2 10)(3 11)(4 12)(5 71)(6 72)(7 69)(8 70)(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)(41 103)(42 104)(43 101)(44 102)(45 107)(46 108)(47 105)(48 106)(49 111)(50 112)(51 109)(52 110)(53 115)(54 116)(55 113)(56 114)(57 119)(58 120)(59 117)(60 118)(61 123)(62 124)(63 121)(64 122)(65 126)(66 127)(67 128)(68 125)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.490C24C4×C4⋊C4C23.63C23C23.65C23C23.81C23C23.83C23C4⋊C4C2×C4C22C22
# reps12633141611

Matrix representation of C23.490C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
410000
310000
000100
001000
000010
000044
,
410000
010000
002000
000200
000040
000011
,
400000
040000
004000
000100
000043
000011
,
320000
020000
003000
000300
000030
000003

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

C23.490C24 in GAP, Magma, Sage, TeX 

C_2^3._{490}C_2^4
 
% in TeX
 
G:=Group("C2^3.490C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1322);
 
// by ID
 
G=gap.SmallGroup(128,1322);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,680,758,723,352,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a,e^2=c*b=b*c,f^2=c,g^2=b*a=a*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,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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