direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.67C23, C24.646C23, C23.176C24, (C22×C4)⋊19Q8, (C22×Q8)⋊16C4, (Q8×C23).4C2, C22.31(C4×Q8), (C22×C4).596D4, C23.829(C2×D4), C22.41(C4⋊Q8), C23.143(C2×Q8), (C22×C42).23C2, C22.67(C23×C4), C22.74(C22×D4), C23.363(C4○D4), C22.24(C22×Q8), C23.285(C22×C4), (C22×C4).450C23, (C23×C4).282C22, C22.89(C22⋊Q8), (C2×C42).1089C22, C22.75(C4.4D4), (C22×Q8).389C22, C2.C42.511C22, C2.6(C2×C4×Q8), (C2×C4)⋊7(C2×Q8), C2.3(C2×C4⋊Q8), (C2×Q8)⋊31(C2×C4), C2.5(C2×C22⋊Q8), C4.61(C2×C22⋊C4), C2.4(C2×C4.4D4), (C2×C4).1389(C2×D4), (C22×C4⋊C4).23C2, C22.68(C2×C4○D4), (C2×C4⋊C4).791C22, (C2×C4).209(C22×C4), (C22×C4).295(C2×C4), C2.10(C22×C22⋊C4), (C2×C4).276(C22⋊C4), C22.133(C2×C22⋊C4), (C2×C2.C42).18C2, SmallGroup(128,1026)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.67C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=bcd, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
Subgroups: 716 in 464 conjugacy classes, 236 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C22×Q8, C22×Q8, C2×C2.C42, C23.67C23, C22×C42, C22×C4⋊C4, Q8×C23, C2×C23.67C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C23.67C23, C22×C22⋊C4, C2×C4×Q8, C2×C22⋊Q8, C2×C4.4D4, C2×C4⋊Q8, C2×C23.67C23
►Regular action on 128 points
Generators in S128
(1 45)(2 46)(3 47)(4 48)(5 111)(6 112)(7 109)(8 110)(9 33)(10 34)(11 35)(12 36)(13 124)(14 121)(15 122)(16 123)(17 107)(18 108)(19 105)(20 106)(21 127)(22 128)(23 125)(24 126)(25 99)(26 100)(27 97)(28 98)(29 119)(30 120)(31 117)(32 118)(37 87)(38 88)(39 85)(40 86)(41 115)(42 116)(43 113)(44 114)(49 83)(50 84)(51 81)(52 82)(53 79)(54 80)(55 77)(56 78)(57 71)(58 72)(59 69)(60 70)(61 95)(62 96)(63 93)(64 94)(65 104)(66 101)(67 102)(68 103)(73 89)(74 90)(75 91)(76 92)
(1 27)(2 28)(3 25)(4 26)(5 69)(6 70)(7 71)(8 72)(9 66)(10 67)(11 68)(12 65)(13 78)(14 79)(15 80)(16 77)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(29 81)(30 82)(31 83)(32 84)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(73 127)(74 128)(75 125)(76 126)
(1 103)(2 104)(3 101)(4 102)(5 37)(6 38)(7 39)(8 40)(9 99)(10 100)(11 97)(12 98)(13 76)(14 73)(15 74)(16 75)(17 57)(18 58)(19 59)(20 60)(21 53)(22 54)(23 55)(24 56)(25 33)(26 34)(27 35)(28 36)(29 61)(30 62)(31 63)(32 64)(41 49)(42 50)(43 51)(44 52)(45 68)(46 65)(47 66)(48 67)(69 105)(70 106)(71 107)(72 108)(77 125)(78 126)(79 127)(80 128)(81 113)(82 114)(83 115)(84 116)(85 109)(86 110)(87 111)(88 112)(89 121)(90 122)(91 123)(92 124)(93 117)(94 118)(95 119)(96 120)
(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 61 33 83)(2 114 34 32)(3 63 35 81)(4 116 36 30)(5 127 107 16)(6 74 108 78)(7 125 105 14)(8 76 106 80)(9 49 45 95)(10 118 46 44)(11 51 47 93)(12 120 48 42)(13 70 128 40)(15 72 126 38)(17 123 111 21)(18 56 112 90)(19 121 109 23)(20 54 110 92)(22 86 124 60)(24 88 122 58)(25 115 103 29)(26 64 104 82)(27 113 101 31)(28 62 102 84)(37 79 71 75)(39 77 69 73)(41 68 119 99)(43 66 117 97)(50 98 96 67)(52 100 94 65)(53 57 91 87)(55 59 89 85)
(1 107 103 71)(2 72 104 108)(3 105 101 69)(4 70 102 106)(5 25 37 33)(6 34 38 26)(7 27 39 35)(8 36 40 28)(9 111 99 87)(10 88 100 112)(11 109 97 85)(12 86 98 110)(13 62 76 30)(14 31 73 63)(15 64 74 32)(16 29 75 61)(17 68 57 45)(18 46 58 65)(19 66 59 47)(20 48 60 67)(21 41 53 49)(22 50 54 42)(23 43 55 51)(24 52 56 44)(77 81 125 113)(78 114 126 82)(79 83 127 115)(80 116 128 84)(89 93 121 117)(90 118 122 94)(91 95 123 119)(92 120 124 96)
G:=sub<Sym(128)| (1,45)(2,46)(3,47)(4,48)(5,111)(6,112)(7,109)(8,110)(9,33)(10,34)(11,35)(12,36)(13,124)(14,121)(15,122)(16,123)(17,107)(18,108)(19,105)(20,106)(21,127)(22,128)(23,125)(24,126)(25,99)(26,100)(27,97)(28,98)(29,119)(30,120)(31,117)(32,118)(37,87)(38,88)(39,85)(40,86)(41,115)(42,116)(43,113)(44,114)(49,83)(50,84)(51,81)(52,82)(53,79)(54,80)(55,77)(56,78)(57,71)(58,72)(59,69)(60,70)(61,95)(62,96)(63,93)(64,94)(65,104)(66,101)(67,102)(68,103)(73,89)(74,90)(75,91)(76,92), (1,27)(2,28)(3,25)(4,26)(5,69)(6,70)(7,71)(8,72)(9,66)(10,67)(11,68)(12,65)(13,78)(14,79)(15,80)(16,77)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(29,81)(30,82)(31,83)(32,84)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(73,127)(74,128)(75,125)(76,126), (1,103)(2,104)(3,101)(4,102)(5,37)(6,38)(7,39)(8,40)(9,99)(10,100)(11,97)(12,98)(13,76)(14,73)(15,74)(16,75)(17,57)(18,58)(19,59)(20,60)(21,53)(22,54)(23,55)(24,56)(25,33)(26,34)(27,35)(28,36)(29,61)(30,62)(31,63)(32,64)(41,49)(42,50)(43,51)(44,52)(45,68)(46,65)(47,66)(48,67)(69,105)(70,106)(71,107)(72,108)(77,125)(78,126)(79,127)(80,128)(81,113)(82,114)(83,115)(84,116)(85,109)(86,110)(87,111)(88,112)(89,121)(90,122)(91,123)(92,124)(93,117)(94,118)(95,119)(96,120), (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), (1,61,33,83)(2,114,34,32)(3,63,35,81)(4,116,36,30)(5,127,107,16)(6,74,108,78)(7,125,105,14)(8,76,106,80)(9,49,45,95)(10,118,46,44)(11,51,47,93)(12,120,48,42)(13,70,128,40)(15,72,126,38)(17,123,111,21)(18,56,112,90)(19,121,109,23)(20,54,110,92)(22,86,124,60)(24,88,122,58)(25,115,103,29)(26,64,104,82)(27,113,101,31)(28,62,102,84)(37,79,71,75)(39,77,69,73)(41,68,119,99)(43,66,117,97)(50,98,96,67)(52,100,94,65)(53,57,91,87)(55,59,89,85), (1,107,103,71)(2,72,104,108)(3,105,101,69)(4,70,102,106)(5,25,37,33)(6,34,38,26)(7,27,39,35)(8,36,40,28)(9,111,99,87)(10,88,100,112)(11,109,97,85)(12,86,98,110)(13,62,76,30)(14,31,73,63)(15,64,74,32)(16,29,75,61)(17,68,57,45)(18,46,58,65)(19,66,59,47)(20,48,60,67)(21,41,53,49)(22,50,54,42)(23,43,55,51)(24,52,56,44)(77,81,125,113)(78,114,126,82)(79,83,127,115)(80,116,128,84)(89,93,121,117)(90,118,122,94)(91,95,123,119)(92,120,124,96)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,111)(6,112)(7,109)(8,110)(9,33)(10,34)(11,35)(12,36)(13,124)(14,121)(15,122)(16,123)(17,107)(18,108)(19,105)(20,106)(21,127)(22,128)(23,125)(24,126)(25,99)(26,100)(27,97)(28,98)(29,119)(30,120)(31,117)(32,118)(37,87)(38,88)(39,85)(40,86)(41,115)(42,116)(43,113)(44,114)(49,83)(50,84)(51,81)(52,82)(53,79)(54,80)(55,77)(56,78)(57,71)(58,72)(59,69)(60,70)(61,95)(62,96)(63,93)(64,94)(65,104)(66,101)(67,102)(68,103)(73,89)(74,90)(75,91)(76,92), (1,27)(2,28)(3,25)(4,26)(5,69)(6,70)(7,71)(8,72)(9,66)(10,67)(11,68)(12,65)(13,78)(14,79)(15,80)(16,77)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(29,81)(30,82)(31,83)(32,84)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(73,127)(74,128)(75,125)(76,126), (1,103)(2,104)(3,101)(4,102)(5,37)(6,38)(7,39)(8,40)(9,99)(10,100)(11,97)(12,98)(13,76)(14,73)(15,74)(16,75)(17,57)(18,58)(19,59)(20,60)(21,53)(22,54)(23,55)(24,56)(25,33)(26,34)(27,35)(28,36)(29,61)(30,62)(31,63)(32,64)(41,49)(42,50)(43,51)(44,52)(45,68)(46,65)(47,66)(48,67)(69,105)(70,106)(71,107)(72,108)(77,125)(78,126)(79,127)(80,128)(81,113)(82,114)(83,115)(84,116)(85,109)(86,110)(87,111)(88,112)(89,121)(90,122)(91,123)(92,124)(93,117)(94,118)(95,119)(96,120), (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), (1,61,33,83)(2,114,34,32)(3,63,35,81)(4,116,36,30)(5,127,107,16)(6,74,108,78)(7,125,105,14)(8,76,106,80)(9,49,45,95)(10,118,46,44)(11,51,47,93)(12,120,48,42)(13,70,128,40)(15,72,126,38)(17,123,111,21)(18,56,112,90)(19,121,109,23)(20,54,110,92)(22,86,124,60)(24,88,122,58)(25,115,103,29)(26,64,104,82)(27,113,101,31)(28,62,102,84)(37,79,71,75)(39,77,69,73)(41,68,119,99)(43,66,117,97)(50,98,96,67)(52,100,94,65)(53,57,91,87)(55,59,89,85), (1,107,103,71)(2,72,104,108)(3,105,101,69)(4,70,102,106)(5,25,37,33)(6,34,38,26)(7,27,39,35)(8,36,40,28)(9,111,99,87)(10,88,100,112)(11,109,97,85)(12,86,98,110)(13,62,76,30)(14,31,73,63)(15,64,74,32)(16,29,75,61)(17,68,57,45)(18,46,58,65)(19,66,59,47)(20,48,60,67)(21,41,53,49)(22,50,54,42)(23,43,55,51)(24,52,56,44)(77,81,125,113)(78,114,126,82)(79,83,127,115)(80,116,128,84)(89,93,121,117)(90,118,122,94)(91,95,123,119)(92,120,124,96) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,111),(6,112),(7,109),(8,110),(9,33),(10,34),(11,35),(12,36),(13,124),(14,121),(15,122),(16,123),(17,107),(18,108),(19,105),(20,106),(21,127),(22,128),(23,125),(24,126),(25,99),(26,100),(27,97),(28,98),(29,119),(30,120),(31,117),(32,118),(37,87),(38,88),(39,85),(40,86),(41,115),(42,116),(43,113),(44,114),(49,83),(50,84),(51,81),(52,82),(53,79),(54,80),(55,77),(56,78),(57,71),(58,72),(59,69),(60,70),(61,95),(62,96),(63,93),(64,94),(65,104),(66,101),(67,102),(68,103),(73,89),(74,90),(75,91),(76,92)], [(1,27),(2,28),(3,25),(4,26),(5,69),(6,70),(7,71),(8,72),(9,66),(10,67),(11,68),(12,65),(13,78),(14,79),(15,80),(16,77),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(29,81),(30,82),(31,83),(32,84),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,116),(73,127),(74,128),(75,125),(76,126)], [(1,103),(2,104),(3,101),(4,102),(5,37),(6,38),(7,39),(8,40),(9,99),(10,100),(11,97),(12,98),(13,76),(14,73),(15,74),(16,75),(17,57),(18,58),(19,59),(20,60),(21,53),(22,54),(23,55),(24,56),(25,33),(26,34),(27,35),(28,36),(29,61),(30,62),(31,63),(32,64),(41,49),(42,50),(43,51),(44,52),(45,68),(46,65),(47,66),(48,67),(69,105),(70,106),(71,107),(72,108),(77,125),(78,126),(79,127),(80,128),(81,113),(82,114),(83,115),(84,116),(85,109),(86,110),(87,111),(88,112),(89,121),(90,122),(91,123),(92,124),(93,117),(94,118),(95,119),(96,120)], [(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)], [(1,61,33,83),(2,114,34,32),(3,63,35,81),(4,116,36,30),(5,127,107,16),(6,74,108,78),(7,125,105,14),(8,76,106,80),(9,49,45,95),(10,118,46,44),(11,51,47,93),(12,120,48,42),(13,70,128,40),(15,72,126,38),(17,123,111,21),(18,56,112,90),(19,121,109,23),(20,54,110,92),(22,86,124,60),(24,88,122,58),(25,115,103,29),(26,64,104,82),(27,113,101,31),(28,62,102,84),(37,79,71,75),(39,77,69,73),(41,68,119,99),(43,66,117,97),(50,98,96,67),(52,100,94,65),(53,57,91,87),(55,59,89,85)], [(1,107,103,71),(2,72,104,108),(3,105,101,69),(4,70,102,106),(5,25,37,33),(6,34,38,26),(7,27,39,35),(8,36,40,28),(9,111,99,87),(10,88,100,112),(11,109,97,85),(12,86,98,110),(13,62,76,30),(14,31,73,63),(15,64,74,32),(16,29,75,61),(17,68,57,45),(18,46,58,65),(19,66,59,47),(20,48,60,67),(21,41,53,49),(22,50,54,42),(23,43,55,51),(24,52,56,44),(77,81,125,113),(78,114,126,82),(79,83,127,115),(80,116,128,84),(89,93,121,117),(90,118,122,94),(91,95,123,119),(92,120,124,96)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 |
| kernel | C2×C23.67C23 | C2×C2.C42 | C23.67C23 | C22×C42 | C22×C4⋊C4 | Q8×C23 | C22×Q8 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 4 | 8 | 1 | 1 | 1 | 16 | 8 | 8 | 8 |
Matrix representation of C2×C23.67C23 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 1 |
| 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 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,4,0,0,0,0,3,3],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,2] >;
C2×C23.67C23 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{67}C_2^3 % in TeX
G:=Group("C2xC2^3.67C2^3"); // GroupNames label
G:=SmallGroup(128,1026);
// by ID
G=gap.SmallGroup(128,1026);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,792,758,184]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b*c*d,g^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,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,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations