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G = C23.407C24  order 128 = 27

124th central stem extension by C23 of C24

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

Aliases: C23.407C24, C22.2022+ 1+4, C2.4Q82, C4⋊C4.20Q8, C42⋊9C4.26C2, C2.23(D4⋊3Q8), C4.29(C42.C2), C22.90(C22×Q8), (C2×C42).527C22, (C22×C4).829C23, C23.81C23.8C2, C2.C42.158C22, C23.63C23.18C2, C23.65C23.44C2, C2.21(C23.37C23), C2.38(C22.47C24), C2.18(C22.34C24), (C4×C4⋊C4).53C2, (C2×C4).43(C2×Q8), C2.12(C2×C42.C2), (C2×C4).129(C4○D4), (C2×C4⋊C4).273C22, C22.284(C2×C4○D4), SmallGroup(128,1239)

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

Derived series
C1 — C23 — C23.407C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.407C24
Lower central
C1 — C23 — C23.407C24
Upper central
C1 — C23 — C23.407C24
Jennings
C1 — C23 — C23.407C24

Generators and relations for C23.407C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=f2=b, g2=a, ab=ba, 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: 324 in 198 conjugacy classes, 112 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×C4⋊C4, C42⋊9C4, C23.63C23, C23.65C23, C23.65C23, C23.81C23, C23.407C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C42.C2, C23.37C23, C22.34C24, C22.47C24, D4⋊3Q8, Q82, C23.407C24

Smallest permutation representation of C23.407C24
►Regular action on 128 points
Generators in S128
(1 77)(2 78)(3 79)(4 80)(5 127)(6 128)(7 125)(8 126)(9 39)(10 40)(11 37)(12 38)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)(29 36)(30 33)(31 34)(32 35)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(65 70)(66 71)(67 72)(68 69)(73 97)(74 98)(75 99)(76 100)(81 86)(82 87)(83 88)(84 85)(89 94)(90 95)(91 96)(92 93)(101 108)(102 105)(103 106)(104 107)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)

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

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

(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)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111224
type++++++-+
imageC1C2C2C2C2C2Q8C4○D42+ 1+4
kernelC23.407C24C4×C4⋊C4C42⋊9C4C23.63C23C23.65C23C23.81C23C4⋊C4C2×C4C22
# reps1212648122

Matrix representation of C23.407C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
330000
020000
000100
001000
000030
000003
,
200000
130000
003000
000300
000002
000030
,
200000
130000
000100
004000
000001
000010
,
300000
420000
004000
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],[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],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,4,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

C23.407C24 in GAP, Magma, Sage, TeX 

C_2^3._{407}C_2^4
 
% in TeX
 
G:=Group("C2^3.407C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1239);
 
// by ID
 
G=gap.SmallGroup(128,1239);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=f^2=b,g^2=a,a*b=b*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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