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G = C23.411C24  order 128 = 27

128th central stem extension by C23 of C24

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

Aliases: C23.411C24, C22.2062+ 1+4, C22.1562- 1+4, C4⋊C4⋊15Q8, C42⋊9C4.27C2, C2.13(Q8⋊3Q8), C2.26(D4⋊3Q8), (C2×C42).51C22, (C22×C4).82C23, C4.41(C42⋊2C2), C22.93(C22×Q8), (C22×Q8).121C22, C23.83C23.8C2, C23.63C23.21C2, C23.65C23.47C2, C23.67C23.35C2, C2.C42.161C22, C2.34(C22.36C24), C2.46(C22.46C24), C2.12(C22.53C24), C2.22(C23.37C23), (C4×C4⋊C4).56C2, (C2×C4).46(C2×Q8), (C2×C4).132(C4○D4), (C2×C4⋊C4).277C22, C2.18(C2×C42⋊2C2), C22.288(C2×C4○D4), SmallGroup(128,1243)

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

Derived series
C1 — C23 — C23.411C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.411C24
Lower central
C1 — C23 — C23.411C24
Upper central
C1 — C23 — C23.411C24
Jennings
C1 — C23 — C23.411C24

Generators and relations for C23.411C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=cb=bc, e2=f2=b, g2=a, ab=ba, 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: 324 in 194 conjugacy classes, 104 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, C4×C4⋊C4, C42⋊9C4, C23.63C23, C23.65C23, C23.67C23, C23.83C23, C23.411C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42⋊2C2, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42⋊2C2, C23.37C23, C22.36C24, C22.46C24, D4⋊3Q8, Q8⋊3Q8, C22.53C24, C23.411C24

Smallest permutation representation of C23.411C24
►Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 37)(4 38)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(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)(41 101)(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)(65 125)(66 126)(67 127)(68 128)

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111112244
type+++++++-+-
imageC1C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.411C24C4×C4⋊C4C42⋊9C4C23.63C23C23.65C23C23.67C23C23.83C23C4⋊C4C2×C4C22C22
# reps121413441611

Matrix representation of C23.411C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
100000
010000
004000
000400
000010
000001
,
340000
020000
002100
002300
000003
000030
,
300000
030000
001300
001400
000001
000040
,
200000
230000
003000
003200
000004
000010
,
400000
040000
001000
000100
000001
000040

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

C23.411C24 in GAP, Magma, Sage, TeX 

C_2^3._{411}C_2^4
 
% in TeX
 
G:=Group("C2^3.411C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1243);
 
// by ID
 
G=gap.SmallGroup(128,1243);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=f^2=b,g^2=a,a*b=b*a,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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