Copied to
clipboard

## G = C2×C23.84C23order 128 = 27

### Direct product of C2 and C23.84C23

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

Aliases: C2×C23.84C23, C24.652C23, C23.300C24, (C23×C4).70C22, C23.373(C4○D4), (C22×C4).500C23, C22.35(C422C2), C2.C42.487C22, C2.8(C2×C422C2), C22.180(C2×C4○D4), (C2×C2.C42).10C2, SmallGroup(128,1132)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23.84C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C2×C23.84C23
 Lower central C1 — C23 — C2×C23.84C23
 Upper central C1 — C24 — C2×C23.84C23
 Jennings C1 — C23 — C2×C23.84C23

Generators and relations for C2×C23.84C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Subgroups: 468 in 258 conjugacy classes, 132 normal (4 characteristic)
C1, C2, C4, C22, C2×C4, C23, C23, C22×C4, C22×C4, C24, C2.C42, C23×C4, C2×C2.C42, C23.84C23, C2×C23.84C23
Quotients: C1, C2, C22, C23, C4○D4, C24, C422C2, C2×C4○D4, C23.84C23, C2×C422C2, C2×C23.84C23

Smallest permutation representation of C2×C23.84C23
Regular action on 128 points
Generators in S128
(1 106)(2 107)(3 108)(4 105)(5 79)(6 80)(7 77)(8 78)(9 28)(10 25)(11 26)(12 27)(13 66)(14 67)(15 68)(16 65)(17 70)(18 71)(19 72)(20 69)(21 92)(22 89)(23 90)(24 91)(29 84)(30 81)(31 82)(32 83)(33 102)(34 103)(35 104)(36 101)(37 95)(38 96)(39 93)(40 94)(41 86)(42 87)(43 88)(44 85)(45 100)(46 97)(47 98)(48 99)(49 118)(50 119)(51 120)(52 117)(53 124)(54 121)(55 122)(56 123)(57 110)(58 111)(59 112)(60 109)(61 116)(62 113)(63 114)(64 115)(73 126)(74 127)(75 128)(76 125)
(1 36)(2 33)(3 34)(4 35)(5 125)(6 126)(7 127)(8 128)(9 13)(10 14)(11 15)(12 16)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 67)(26 68)(27 65)(28 66)(29 69)(30 70)(31 71)(32 72)(37 98)(38 99)(39 100)(40 97)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)(73 80)(74 77)(75 78)(76 79)(101 106)(102 107)(103 108)(104 105)(109 114)(110 115)(111 116)(112 113)(117 122)(118 123)(119 124)(120 121)
(1 68)(2 65)(3 66)(4 67)(5 47)(6 48)(7 45)(8 46)(9 103)(10 104)(11 101)(12 102)(13 108)(14 105)(15 106)(16 107)(17 59)(18 60)(19 57)(20 58)(21 124)(22 121)(23 122)(24 123)(25 35)(26 36)(27 33)(28 34)(29 116)(30 113)(31 114)(32 115)(37 76)(38 73)(39 74)(40 75)(41 51)(42 52)(43 49)(44 50)(53 92)(54 89)(55 90)(56 91)(61 84)(62 81)(63 82)(64 83)(69 111)(70 112)(71 109)(72 110)(77 100)(78 97)(79 98)(80 99)(85 119)(86 120)(87 117)(88 118)(93 127)(94 128)(95 125)(96 126)
(1 28)(2 25)(3 26)(4 27)(5 93)(6 94)(7 95)(8 96)(9 106)(10 107)(11 108)(12 105)(13 101)(14 102)(15 103)(16 104)(17 64)(18 61)(19 62)(20 63)(21 117)(22 118)(23 119)(24 120)(29 109)(30 110)(31 111)(32 112)(33 67)(34 68)(35 65)(36 66)(37 77)(38 78)(39 79)(40 80)(41 56)(42 53)(43 54)(44 55)(45 125)(46 126)(47 127)(48 128)(49 89)(50 90)(51 91)(52 92)(57 81)(58 82)(59 83)(60 84)(69 114)(70 115)(71 116)(72 113)(73 97)(74 98)(75 99)(76 100)(85 122)(86 123)(87 124)(88 121)
(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 26 124)(2 22 27 120)(3 87 28 122)(4 24 25 118)(5 61 95 20)(6 59 96 81)(7 63 93 18)(8 57 94 83)(9 55 108 42)(10 49 105 91)(11 53 106 44)(12 51 107 89)(13 52 103 90)(14 56 104 43)(15 50 101 92)(16 54 102 41)(17 126 62 48)(19 128 64 46)(21 68 119 36)(23 66 117 34)(29 76 111 98)(30 80 112 38)(31 74 109 100)(32 78 110 40)(33 86 65 121)(35 88 67 123)(37 69 79 116)(39 71 77 114)(45 82 127 60)(47 84 125 58)(70 73 113 99)(72 75 115 97)
(1 111 36 116)(2 70 33 30)(3 109 34 114)(4 72 35 32)(5 42 125 90)(6 49 126 56)(7 44 127 92)(8 51 128 54)(9 82 13 18)(10 64 14 57)(11 84 15 20)(12 62 16 59)(17 102 81 107)(19 104 83 105)(21 77 85 74)(22 97 86 40)(23 79 87 76)(24 99 88 38)(25 115 67 110)(26 29 68 69)(27 113 65 112)(28 31 66 71)(37 122 98 117)(39 124 100 119)(41 94 89 46)(43 96 91 48)(45 50 93 53)(47 52 95 55)(58 101 61 106)(60 103 63 108)(73 123 80 118)(75 121 78 120)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4AB order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4

44 irreducible representations

 dim 1 1 1 2 type + + + image C1 C2 C2 C4○D4 kernel C2×C23.84C23 C2×C2.C42 C23.84C23 C23 # reps 1 7 8 28

Matrix representation of C2×C23.84C23 in GL8(𝔽5)

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

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,3,3],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1] >;

C2×C23.84C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{84}C_2^3
% in TeX

G:=Group("C2xC2^3.84C2^3");
// GroupNames label

G:=SmallGroup(128,1132);
// by ID

G=gap.SmallGroup(128,1132);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,100]);
// Polycyclic

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

׿
×
𝔽