p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.362C24, C22.1242- 1+4, C4⋊C4⋊14Q8, C2.16(D4×Q8), C4⋊C4.330D4, C2.9(Q8⋊3Q8), C2.30(D4⋊6D4), C22.81(C22×Q8), (C2×C42).505C22, (C22×C4).815C23, C22.242(C22×D4), (C22×Q8).429C22, C2.34(C22.19C24), C23.81C23.6C2, C23.78C23.4C2, C2.C42.119C22, C23.65C23.42C2, C23.63C23.16C2, C2.23(C22.46C24), C2.12(C22.35C24), C2.17(C23.37C23), (C4×C4⋊C4).51C2, (C2×C4×Q8).32C2, (C2×C4).34(C2×Q8), (C2×C4).338(C2×D4), (C2×C4).366(C4○D4), (C2×C4⋊C4).243C22, C22.239(C2×C4○D4), (C2×C42.C2).17C2, SmallGroup(128,1194)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.362C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=a, g2=b, ab=ba, ac=ca, 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: 356 in 226 conjugacy classes, 108 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C4×Q8, C42.C2, C22×Q8, C4×C4⋊C4, C23.63C23, C23.65C23, C23.78C23, C23.81C23, C2×C4×Q8, C2×C42.C2, C23.362C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C22.19C24, C23.37C23, C22.35C24, D4⋊6D4, D4×Q8, C22.46C24, Q8⋊3Q8, C23.362C24
►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 87 3 85)(2 86 4 88)(5 114 7 116)(6 113 8 115)(9 27 11 25)(10 26 12 28)(13 31 15 29)(14 30 16 32)(17 35 19 33)(18 34 20 36)(21 39 23 37)(22 38 24 40)(41 59 43 57)(42 58 44 60)(45 63 47 61)(46 62 48 64)(49 68 51 66)(50 67 52 65)(53 72 55 70)(54 71 56 69)(73 91 75 89)(74 90 76 92)(77 95 79 93)(78 94 80 96)(81 99 83 97)(82 98 84 100)(101 119 103 117)(102 118 104 120)(105 123 107 121)(106 122 108 124)(109 127 111 125)(110 126 112 128)
(1 79 3 77)(2 20 4 18)(5 32 7 30)(6 89 8 91)(9 19 11 17)(10 80 12 78)(13 23 15 21)(14 84 16 82)(22 74 24 76)(25 66 27 68)(26 128 28 126)(29 70 31 72)(33 59 35 57)(34 118 36 120)(37 63 39 61)(38 122 40 124)(41 51 43 49)(42 110 44 112)(45 55 47 53)(46 114 48 116)(50 104 52 102)(54 108 56 106)(58 94 60 96)(62 98 64 100)(65 86 67 88)(69 90 71 92)(73 83 75 81)(85 127 87 125)(93 117 95 119)(97 121 99 123)(101 111 103 109)(105 115 107 113)
(1 13 9 73)(2 16 10 76)(3 15 11 75)(4 14 12 74)(5 126 71 65)(6 125 72 68)(7 128 69 67)(8 127 70 66)(17 81 77 21)(18 84 78 24)(19 83 79 23)(20 82 80 22)(25 89 85 29)(26 92 86 32)(27 91 87 31)(28 90 88 30)(33 97 93 37)(34 100 94 40)(35 99 95 39)(36 98 96 38)(41 107 103 45)(42 106 104 48)(43 105 101 47)(44 108 102 46)(49 115 111 53)(50 114 112 56)(51 113 109 55)(52 116 110 54)(57 123 119 61)(58 122 120 64)(59 121 117 63)(60 124 118 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), (1,87,3,85)(2,86,4,88)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,59,43,57)(42,58,44,60)(45,63,47,61)(46,62,48,64)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(101,119,103,117)(102,118,104,120)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128), (1,79,3,77)(2,20,4,18)(5,32,7,30)(6,89,8,91)(9,19,11,17)(10,80,12,78)(13,23,15,21)(14,84,16,82)(22,74,24,76)(25,66,27,68)(26,128,28,126)(29,70,31,72)(33,59,35,57)(34,118,36,120)(37,63,39,61)(38,122,40,124)(41,51,43,49)(42,110,44,112)(45,55,47,53)(46,114,48,116)(50,104,52,102)(54,108,56,106)(58,94,60,96)(62,98,64,100)(65,86,67,88)(69,90,71,92)(73,83,75,81)(85,127,87,125)(93,117,95,119)(97,121,99,123)(101,111,103,109)(105,115,107,113), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62)>;
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), (1,87,3,85)(2,86,4,88)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,59,43,57)(42,58,44,60)(45,63,47,61)(46,62,48,64)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(101,119,103,117)(102,118,104,120)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128), (1,79,3,77)(2,20,4,18)(5,32,7,30)(6,89,8,91)(9,19,11,17)(10,80,12,78)(13,23,15,21)(14,84,16,82)(22,74,24,76)(25,66,27,68)(26,128,28,126)(29,70,31,72)(33,59,35,57)(34,118,36,120)(37,63,39,61)(38,122,40,124)(41,51,43,49)(42,110,44,112)(45,55,47,53)(46,114,48,116)(50,104,52,102)(54,108,56,106)(58,94,60,96)(62,98,64,100)(65,86,67,88)(69,90,71,92)(73,83,75,81)(85,127,87,125)(93,117,95,119)(97,121,99,123)(101,111,103,109)(105,115,107,113), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62) );
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)], [(1,87,3,85),(2,86,4,88),(5,114,7,116),(6,113,8,115),(9,27,11,25),(10,26,12,28),(13,31,15,29),(14,30,16,32),(17,35,19,33),(18,34,20,36),(21,39,23,37),(22,38,24,40),(41,59,43,57),(42,58,44,60),(45,63,47,61),(46,62,48,64),(49,68,51,66),(50,67,52,65),(53,72,55,70),(54,71,56,69),(73,91,75,89),(74,90,76,92),(77,95,79,93),(78,94,80,96),(81,99,83,97),(82,98,84,100),(101,119,103,117),(102,118,104,120),(105,123,107,121),(106,122,108,124),(109,127,111,125),(110,126,112,128)], [(1,79,3,77),(2,20,4,18),(5,32,7,30),(6,89,8,91),(9,19,11,17),(10,80,12,78),(13,23,15,21),(14,84,16,82),(22,74,24,76),(25,66,27,68),(26,128,28,126),(29,70,31,72),(33,59,35,57),(34,118,36,120),(37,63,39,61),(38,122,40,124),(41,51,43,49),(42,110,44,112),(45,55,47,53),(46,114,48,116),(50,104,52,102),(54,108,56,106),(58,94,60,96),(62,98,64,100),(65,86,67,88),(69,90,71,92),(73,83,75,81),(85,127,87,125),(93,117,95,119),(97,121,99,123),(101,111,103,109),(105,115,107,113)], [(1,13,9,73),(2,16,10,76),(3,15,11,75),(4,14,12,74),(5,126,71,65),(6,125,72,68),(7,128,69,67),(8,127,70,66),(17,81,77,21),(18,84,78,24),(19,83,79,23),(20,82,80,22),(25,89,85,29),(26,92,86,32),(27,91,87,31),(28,90,88,30),(33,97,93,37),(34,100,94,40),(35,99,95,39),(36,98,96,38),(41,107,103,45),(42,106,104,48),(43,105,101,47),(44,108,102,46),(49,115,111,53),(50,114,112,56),(51,113,109,55),(52,116,110,54),(57,123,119,61),(58,122,120,64),(59,121,117,63),(60,124,118,62)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2- 1+4 |
| kernel | C23.362C24 | C4×C4⋊C4 | C23.63C23 | C23.65C23 | C23.78C23 | C23.81C23 | C2×C4×Q8 | C2×C42.C2 | C4⋊C4 | C4⋊C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C23.362C24 ►in GL6(𝔽5)
| 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 |
| 4 | 4 | 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 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
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],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
C23.362C24 in GAP, Magma, Sage, TeX
C_2^3._{362}C_2^4 % in TeX
G:=Group("C2^3.362C2^4"); // GroupNames label
G:=SmallGroup(128,1194);
// by ID
G=gap.SmallGroup(128,1194);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=a,g^2=b,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,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