p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.218C24, C22.392- 1+4, C22.562+ 1+4, C42.C2⋊14C4, C42⋊5C4.4C2, C42.183(C2×C4), (C2×C42).422C22, (C22×C4).483C23, C22.109(C23×C4), C2.C42.53C22, C23.63C23.5C2, C23.65C23.32C2, C2.9(C22.33C24), C2.7(C22.35C24), C2.20(C23.33C23), C2.12(C23.32C23), (C4×C4⋊C4).37C2, C2.21(C4×C4○D4), C4⋊C4.107(C2×C4), (C2×C4).38(C22×C4), (C2×C4).520(C4○D4), (C2×C4⋊C4).813C22, C22.103(C2×C4○D4), (C2×C42.C2).12C2, SmallGroup(128,1068)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.218C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=a, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, fef-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, eg=ge, fg=gf >
Subgroups: 316 in 214 conjugacy classes, 136 normal (22 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×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C4×C4⋊C4, C4×C4⋊C4, C42⋊5C4, C23.63C23, C23.65C23, C2×C42.C2, C23.218C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, 2+ 1+4, 2- 1+4, C4×C4○D4, C23.32C23, C23.33C23, C22.33C24, C22.35C24, C23.218C24
►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 85 3 87)(2 26 4 28)(5 82 7 84)(6 23 8 21)(9 27 11 25)(10 88 12 86)(13 31 15 29)(14 92 16 90)(17 35 19 33)(18 96 20 94)(22 72 24 70)(30 74 32 76)(34 78 36 80)(37 55 39 53)(38 116 40 114)(41 59 43 57)(42 120 44 118)(45 63 47 61)(46 124 48 122)(49 65 51 67)(50 128 52 126)(54 98 56 100)(58 102 60 104)(62 106 64 108)(66 112 68 110)(69 81 71 83)(73 91 75 89)(77 95 79 93)(97 115 99 113)(101 119 103 117)(105 123 107 121)(109 127 111 125)
(1 19 11 79)(2 48 12 108)(3 17 9 77)(4 46 10 106)(5 118 70 58)(6 91 71 31)(7 120 72 60)(8 89 69 29)(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 33 85 93)(26 62 86 122)(27 35 87 95)(28 64 88 124)(30 68 90 126)(32 66 92 128)(34 114 94 54)(36 116 96 56)(37 105 97 45)(39 107 99 47)(41 109 101 49)(43 111 103 51)(53 61 113 121)(55 63 115 123)(57 67 117 125)(59 65 119 127)
(1 43 39 75)(2 104 40 16)(3 41 37 73)(4 102 38 14)(5 94 126 62)(6 35 127 123)(7 96 128 64)(8 33 125 121)(9 101 97 13)(10 42 98 74)(11 103 99 15)(12 44 100 76)(17 109 105 21)(18 50 106 82)(19 111 107 23)(20 52 108 84)(22 78 110 46)(24 80 112 48)(25 117 113 29)(26 58 114 90)(27 119 115 31)(28 60 116 92)(30 86 118 54)(32 88 120 56)(34 68 122 70)(36 66 124 72)(45 81 77 49)(47 83 79 51)(53 89 85 57)(55 91 87 59)(61 69 93 67)(63 71 95 65)
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), (1,85,3,87)(2,26,4,28)(5,82,7,84)(6,23,8,21)(9,27,11,25)(10,88,12,86)(13,31,15,29)(14,92,16,90)(17,35,19,33)(18,96,20,94)(22,72,24,70)(30,74,32,76)(34,78,36,80)(37,55,39,53)(38,116,40,114)(41,59,43,57)(42,120,44,118)(45,63,47,61)(46,124,48,122)(49,65,51,67)(50,128,52,126)(54,98,56,100)(58,102,60,104)(62,106,64,108)(66,112,68,110)(69,81,71,83)(73,91,75,89)(77,95,79,93)(97,115,99,113)(101,119,103,117)(105,123,107,121)(109,127,111,125), (1,19,11,79)(2,48,12,108)(3,17,9,77)(4,46,10,106)(5,118,70,58)(6,91,71,31)(7,120,72,60)(8,89,69,29)(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)(25,33,85,93)(26,62,86,122)(27,35,87,95)(28,64,88,124)(30,68,90,126)(32,66,92,128)(34,114,94,54)(36,116,96,56)(37,105,97,45)(39,107,99,47)(41,109,101,49)(43,111,103,51)(53,61,113,121)(55,63,115,123)(57,67,117,125)(59,65,119,127), (1,43,39,75)(2,104,40,16)(3,41,37,73)(4,102,38,14)(5,94,126,62)(6,35,127,123)(7,96,128,64)(8,33,125,121)(9,101,97,13)(10,42,98,74)(11,103,99,15)(12,44,100,76)(17,109,105,21)(18,50,106,82)(19,111,107,23)(20,52,108,84)(22,78,110,46)(24,80,112,48)(25,117,113,29)(26,58,114,90)(27,119,115,31)(28,60,116,92)(30,86,118,54)(32,88,120,56)(34,68,122,70)(36,66,124,72)(45,81,77,49)(47,83,79,51)(53,89,85,57)(55,91,87,59)(61,69,93,67)(63,71,95,65)>;
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), (1,85,3,87)(2,26,4,28)(5,82,7,84)(6,23,8,21)(9,27,11,25)(10,88,12,86)(13,31,15,29)(14,92,16,90)(17,35,19,33)(18,96,20,94)(22,72,24,70)(30,74,32,76)(34,78,36,80)(37,55,39,53)(38,116,40,114)(41,59,43,57)(42,120,44,118)(45,63,47,61)(46,124,48,122)(49,65,51,67)(50,128,52,126)(54,98,56,100)(58,102,60,104)(62,106,64,108)(66,112,68,110)(69,81,71,83)(73,91,75,89)(77,95,79,93)(97,115,99,113)(101,119,103,117)(105,123,107,121)(109,127,111,125), (1,19,11,79)(2,48,12,108)(3,17,9,77)(4,46,10,106)(5,118,70,58)(6,91,71,31)(7,120,72,60)(8,89,69,29)(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)(25,33,85,93)(26,62,86,122)(27,35,87,95)(28,64,88,124)(30,68,90,126)(32,66,92,128)(34,114,94,54)(36,116,96,56)(37,105,97,45)(39,107,99,47)(41,109,101,49)(43,111,103,51)(53,61,113,121)(55,63,115,123)(57,67,117,125)(59,65,119,127), (1,43,39,75)(2,104,40,16)(3,41,37,73)(4,102,38,14)(5,94,126,62)(6,35,127,123)(7,96,128,64)(8,33,125,121)(9,101,97,13)(10,42,98,74)(11,103,99,15)(12,44,100,76)(17,109,105,21)(18,50,106,82)(19,111,107,23)(20,52,108,84)(22,78,110,46)(24,80,112,48)(25,117,113,29)(26,58,114,90)(27,119,115,31)(28,60,116,92)(30,86,118,54)(32,88,120,56)(34,68,122,70)(36,66,124,72)(45,81,77,49)(47,83,79,51)(53,89,85,57)(55,91,87,59)(61,69,93,67)(63,71,95,65) );
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)], [(1,85,3,87),(2,26,4,28),(5,82,7,84),(6,23,8,21),(9,27,11,25),(10,88,12,86),(13,31,15,29),(14,92,16,90),(17,35,19,33),(18,96,20,94),(22,72,24,70),(30,74,32,76),(34,78,36,80),(37,55,39,53),(38,116,40,114),(41,59,43,57),(42,120,44,118),(45,63,47,61),(46,124,48,122),(49,65,51,67),(50,128,52,126),(54,98,56,100),(58,102,60,104),(62,106,64,108),(66,112,68,110),(69,81,71,83),(73,91,75,89),(77,95,79,93),(97,115,99,113),(101,119,103,117),(105,123,107,121),(109,127,111,125)], [(1,19,11,79),(2,48,12,108),(3,17,9,77),(4,46,10,106),(5,118,70,58),(6,91,71,31),(7,120,72,60),(8,89,69,29),(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),(25,33,85,93),(26,62,86,122),(27,35,87,95),(28,64,88,124),(30,68,90,126),(32,66,92,128),(34,114,94,54),(36,116,96,56),(37,105,97,45),(39,107,99,47),(41,109,101,49),(43,111,103,51),(53,61,113,121),(55,63,115,123),(57,67,117,125),(59,65,119,127)], [(1,43,39,75),(2,104,40,16),(3,41,37,73),(4,102,38,14),(5,94,126,62),(6,35,127,123),(7,96,128,64),(8,33,125,121),(9,101,97,13),(10,42,98,74),(11,103,99,15),(12,44,100,76),(17,109,105,21),(18,50,106,82),(19,111,107,23),(20,52,108,84),(22,78,110,46),(24,80,112,48),(25,117,113,29),(26,58,114,90),(27,119,115,31),(28,60,116,92),(30,86,118,54),(32,88,120,56),(34,68,122,70),(36,66,124,72),(45,81,77,49),(47,83,79,51),(53,89,85,57),(55,91,87,59),(61,69,93,67),(63,71,95,65)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.218C24 | C4×C4⋊C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C2×C42.C2 | C42.C2 | C2×C4 | C22 | C22 |
| # reps | 1 | 3 | 1 | 8 | 2 | 1 | 16 | 8 | 1 | 3 |
Matrix representation of C23.218C24 ►in GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 1 | 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 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3] >;
C23.218C24 in GAP, Magma, Sage, TeX
C_2^3._{218}C_2^4 % in TeX
G:=Group("C2^3.218C2^4"); // GroupNames label
G:=SmallGroup(128,1068);
// by ID
G=gap.SmallGroup(128,1068);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,100,675,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c,f^2=a,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,f*e*f^-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,e*g=g*e,f*g=g*f>;
// generators/relations