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G = C23.264C24  order 128 = 27

117th central extension by C23 of C24

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

Aliases: C23.264C24, C22.692- 1+4, C22.952+ 1+4, C42.C2⋊17C4, C42.197(C2×C4), C42⋊8C4.25C2, C22.155(C23×C4), (C2×C42).452C22, (C22×C4).492C23, C2.C42.72C22, C23.63C23.12C2, C23.65C23.38C2, C2.5(C22.56C24), C2.1(C22.58C24), C2.5(C22.57C24), C2.42(C23.33C23), C2.22(C23.32C23), C4⋊C4.113(C2×C4), (C2×C4).60(C22×C4), (C2×C4⋊C4).199C22, (C2×C42.C2).15C2, SmallGroup(128,1114)

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

Derived series
C1 — C22 — C23.264C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42.C2 — C23.264C24
Lower central
C1 — C22 — C23.264C24
Upper central
C1 — C23 — C23.264C24
Jennings
C1 — C23 — C23.264C24

Generators and relations for C23.264C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=f2=b, g2=a, ab=ba, 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, ce=ec, cf=fc, cg=gc, gdg-1=abd, fef-1=abe, fg=gf >

Subgroups: 316 in 204 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C42.C2, C42⋊8C4, C23.63C23, C23.65C23, C2×C42.C2, C23.264C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C23.32C23, C23.33C23, C22.56C24, C22.57C24, C22.58C24, C23.264C24

Smallest permutation representation of C23.264C24
►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)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4AD
order12···24···4
size11···14···4

38 irreducible representations

dim11111144
type++++++-
imageC1C2C2C2C2C42+ 1+42- 1+4
kernelC23.264C24C42⋊8C4C23.63C23C23.65C23C2×C42.C2C42.C2C22C22
# reps128411624

Matrix representation of C23.264C24 ►in GL9(𝔽5)

100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
400000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
200000000
041110000
001100000
000400000
000110000
000000010
000002313
000004000
000000002
,
100000000
001220000
040200000
001110000
014340000
000002000
000000300
000000020
000004023
,
100000000
010200000
000110000
000400000
001100000
000000100
000004000
000003242
000000241
,
100000000
042000000
041000000
004010000
041400000
000004000
000000400
000000010
000003201

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,4,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,3,0,2],[1,0,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,1,0,1,4,0,0,0,0,0,2,2,1,3,0,0,0,0,0,2,0,1,4,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,1,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,0,2,2,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,0,0,0,4,4,0,4,0,0,0,0,0,2,1,4,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1] >;
 

C23.264C24 in GAP, Magma, Sage, TeX 

C_2^3._{264}C_2^4
 
% in TeX
 
G:=Group("C2^3.264C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1114);
 
// by ID
 
G=gap.SmallGroup(128,1114);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,555,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=f^2=b,g^2=a,a*b=b*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,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*e*f^-1=a*b*e,f*g=g*f>;
 
// generators/relations
 

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