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## G = C2×C22.58C24order 128 = 27

### Direct product of C2 and C22.58C24

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2×C22.58C24, C22.119C25, C42.583C23, C23.280C24, C22.872- 1+4, C4⋊C4.308C23, (C2×C4).109C24, (C2×C42).963C22, C2.37(C2×2- 1+4), (C22×C4).1217C23, C42.C2.160C22, (C2×C4⋊C4).717C22, (C2×C42.C2).38C2, SmallGroup(128,2262)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.58C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42.C2 — C2×C22.58C24
 Lower central C1 — C22 — C2×C22.58C24
 Upper central C1 — C23 — C2×C22.58C24
 Jennings C1 — C22 — C2×C22.58C24

Generators and relations for C2×C22.58C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=b, e2=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=bcd, fef-1=bce, fg=gf >

Subgroups: 508 in 448 conjugacy classes, 388 normal (4 characteristic)
C1, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2×C42, C2×C4⋊C4, C42.C2, C2×C42.C2, C22.58C24, C2×C22.58C24
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C22.58C24, C2×2- 1+4, C2×C22.58C24

Smallest permutation representation of C2×C22.58C24
Regular action on 128 points
Generators in S128
(1 97)(2 98)(3 99)(4 100)(5 36)(6 33)(7 34)(8 35)(9 103)(10 104)(11 101)(12 102)(13 127)(14 128)(15 125)(16 126)(17 111)(18 112)(19 109)(20 110)(21 107)(22 108)(23 105)(24 106)(25 119)(26 120)(27 117)(28 118)(29 115)(30 116)(31 113)(32 114)(37 123)(38 124)(39 121)(40 122)(41 75)(42 76)(43 73)(44 74)(45 67)(46 68)(47 65)(48 66)(49 83)(50 84)(51 81)(52 82)(53 79)(54 80)(55 77)(56 78)(57 91)(58 92)(59 89)(60 90)(61 87)(62 88)(63 85)(64 86)(69 96)(70 93)(71 94)(72 95)
(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 35)(2 36)(3 33)(4 34)(5 98)(6 99)(7 100)(8 97)(9 45)(10 46)(11 47)(12 48)(13 72)(14 69)(15 70)(16 71)(17 53)(18 54)(19 55)(20 56)(21 41)(22 42)(23 43)(24 44)(25 61)(26 62)(27 63)(28 64)(29 49)(30 50)(31 51)(32 52)(37 57)(38 58)(39 59)(40 60)(65 101)(66 102)(67 103)(68 104)(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)
(1 115 35 83)(2 114 36 82)(3 113 33 81)(4 116 34 84)(5 52 98 32)(6 51 99 31)(7 50 100 30)(8 49 97 29)(9 117 45 85)(10 120 46 88)(11 119 47 87)(12 118 48 86)(13 77 72 109)(14 80 69 112)(15 79 70 111)(16 78 71 110)(17 125 53 93)(18 128 54 96)(19 127 55 95)(20 126 56 94)(21 121 41 89)(22 124 42 92)(23 123 43 91)(24 122 44 90)(25 65 61 101)(26 68 62 104)(27 67 63 103)(28 66 64 102)(37 73 57 105)(38 76 58 108)(39 75 59 107)(40 74 60 106)
(1 107 35 75)(2 76 36 108)(3 105 33 73)(4 74 34 106)(5 22 98 42)(6 43 99 23)(7 24 100 44)(8 41 97 21)(9 109 45 77)(10 78 46 110)(11 111 47 79)(12 80 48 112)(13 119 72 87)(14 88 69 120)(15 117 70 85)(16 86 71 118)(17 65 53 101)(18 102 54 66)(19 67 55 103)(20 104 56 68)(25 95 61 127)(26 128 62 96)(27 93 63 125)(28 126 64 94)(29 91 49 123)(30 124 50 92)(31 89 51 121)(32 122 52 90)(37 115 57 83)(38 84 58 116)(39 113 59 81)(40 82 60 114)
(1 65 3 67)(2 104 4 102)(5 46 7 48)(6 9 8 11)(10 100 12 98)(13 91 15 89)(14 122 16 124)(17 73 19 75)(18 108 20 106)(21 79 23 77)(22 110 24 112)(25 83 27 81)(26 114 28 116)(29 85 31 87)(30 120 32 118)(33 103 35 101)(34 66 36 68)(37 93 39 95)(38 128 40 126)(41 111 43 109)(42 78 44 80)(45 97 47 99)(49 117 51 119)(50 88 52 86)(53 105 55 107)(54 76 56 74)(57 125 59 127)(58 96 60 94)(61 115 63 113)(62 82 64 84)(69 90 71 92)(70 121 72 123)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4AD order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4

38 irreducible representations

 dim 1 1 1 4 type + + + - image C1 C2 C2 2- 1+4 kernel C2×C22.58C24 C2×C42.C2 C22.58C24 C22 # reps 1 15 16 6

Matrix representation of C2×C22.58C24 in GL12(𝔽5)

 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 4 4 2 1 0 0 0 0 0 0 0 0 3 4 4 1 0 0 0 0 0 0 0 0 4 0 1 3 0 0 0 0 0 0 0 0 0 3 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 4 0 0 0 0 0 0 0 0 2 2 0 4 0 0 0 0 0 0 0 0 2 1 2 1 0 0 0 0 0 0 0 0 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 2 3 4 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 2 3
,
 1 0 0 3 0 0 0 0 0 0 0 0 1 4 3 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 4 1 3 3 0 0 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 2 3 4 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3 2 3
,
 0 0 1 0 0 0 0 0 0 0 0 0 2 3 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 0 0 0 0 0 0 0 0 0 0 0 0 4 0 2 0 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 0 0 0 4 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 4 1 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 1 3 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 4 4 0 0 0 0 0 0 0 0 0 1 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 3 1 0 0 0 0 0 0 0 0 0 0 3 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 4 1 3 0 0 0 0 0 0 0 0 1 0 1 4

G:=sub<GL(12,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,3,4,0,0,0,0,0,0,0,0,0,4,4,0,3,0,0,0,0,0,0,0,0,2,4,1,4,0,0,0,0,0,0,0,0,1,1,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,0,0,0,0,0,0,0,0,2,2,1,2,0,0,0,0,0,0,0,0,3,0,2,1,0,0,0,0,0,0,0,0,4,4,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,3,0,2,0,0,0,0,0,0,0,0,0,4,0,3],[1,1,0,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,3,0,4,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,4,4,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,2,0,3,0,0,0,0,0,0,0,0,2,3,0,2,0,0,0,0,0,0,0,0,0,4,0,3],[0,2,4,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,2,4,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,2,2,1,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,3,0,1],[1,0,1,1,0,0,0,0,0,0,0,0,3,4,4,3,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,0,0,2,1,3,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,3,4] >;

C2×C22.58C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{58}C_2^4
% in TeX

G:=Group("C2xC2^2.58C2^4");
// GroupNames label

G:=SmallGroup(128,2262);
// by ID

G=gap.SmallGroup(128,2262);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,1059,352,2915,570,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=b,e^2=f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
// generators/relations

׿
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