p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.631C23, (C22×C4).6Q8, C22.34(C4×Q8), C23.721(C2×D4), (C22×C4).196D4, C22.124(C4×D4), C22.22(C4⋊Q8), (C23×C4).9C22, C23.130(C2×Q8), C22.68C22≀C2, C23.344(C4○D4), C23.300(C22×C4), C22.61(C22⋊Q8), C2.2(C23.Q8), C2.6(C23.7Q8), C22.100(C4⋊D4), C2.4(C23.10D4), C22.50(C4.4D4), C2.8(C23.23D4), C22.19(C42.C2), C22.25(C42⋊2C2), C22.74(C42⋊C2), C2.1(C23.78C23), C2.1(C23.81C23), C2.1(C23.83C23), C2.4(C23.65C23), C2.4(C23.67C23), C2.4(C23.63C23), C22.73(C22.D4), (C2×C4⋊C4)⋊18C4, (C2×C4)⋊3(C4⋊C4), (C22×C4⋊C4).3C2, C22.77(C2×C4⋊C4), (C2×C4).65(C22⋊C4), (C22×C4).169(C2×C4), C22.141(C2×C22⋊C4), (C2×C2.C42).12C2, SmallGroup(128,173)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.631C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=a, ab=ba, ac=ca, ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd >
Subgroups: 508 in 282 conjugacy classes, 116 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C2×C2.C42, C2×C2.C42, C22×C4⋊C4, C24.631C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23.7Q8, C23.23D4, C23.63C23, C23.65C23, C23.67C23, C23.10D4, C23.78C23, C23.Q8, C23.81C23, C23.83C23, C24.631C23
►Regular action on 128 points
Generators in S128
(1 75)(2 76)(3 73)(4 74)(5 34)(6 35)(7 36)(8 33)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(37 41)(38 42)(39 43)(40 44)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 60)(61 67)(62 68)(63 65)(64 66)(69 93)(70 94)(71 95)(72 96)(77 81)(78 82)(79 83)(80 84)(85 89)(86 90)(87 91)(88 92)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(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 91 75 87)(2 88 76 92)(3 89 73 85)(4 86 74 90)(5 18 34 22)(6 23 35 19)(7 20 36 24)(8 21 33 17)(9 29 13 25)(10 26 14 30)(11 31 15 27)(12 28 16 32)(37 57 41 53)(38 54 42 58)(39 59 43 55)(40 56 44 60)(45 67 49 61)(46 62 50 68)(47 65 51 63)(48 64 52 66)(69 81 93 77)(70 78 94 82)(71 83 95 79)(72 80 96 84)(97 117 101 113)(98 114 102 118)(99 119 103 115)(100 116 104 120)(105 125 109 121)(106 122 110 126)(107 127 111 123)(108 124 112 128)
(1 83 75 79)(2 24 76 20)(3 81 73 77)(4 22 74 18)(5 118 34 114)(6 59 35 55)(7 120 36 116)(8 57 33 53)(9 21 13 17)(10 82 14 78)(11 23 15 19)(12 84 16 80)(25 67 29 61)(26 126 30 122)(27 65 31 63)(28 128 32 124)(37 49 41 45)(38 110 42 106)(39 51 43 47)(40 112 44 108)(46 98 50 102)(48 100 52 104)(54 70 58 94)(56 72 60 96)(62 86 68 90)(64 88 66 92)(69 117 93 113)(71 119 95 115)(85 125 89 121)(87 127 91 123)(97 109 101 105)(99 111 103 107)
G:=sub<Sym(128)| (1,75)(2,76)(3,73)(4,74)(5,34)(6,35)(7,36)(8,33)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60)(61,67)(62,68)(63,65)(64,66)(69,93)(70,94)(71,95)(72,96)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,91,75,87)(2,88,76,92)(3,89,73,85)(4,86,74,90)(5,18,34,22)(6,23,35,19)(7,20,36,24)(8,21,33,17)(9,29,13,25)(10,26,14,30)(11,31,15,27)(12,28,16,32)(37,57,41,53)(38,54,42,58)(39,59,43,55)(40,56,44,60)(45,67,49,61)(46,62,50,68)(47,65,51,63)(48,64,52,66)(69,81,93,77)(70,78,94,82)(71,83,95,79)(72,80,96,84)(97,117,101,113)(98,114,102,118)(99,119,103,115)(100,116,104,120)(105,125,109,121)(106,122,110,126)(107,127,111,123)(108,124,112,128), (1,83,75,79)(2,24,76,20)(3,81,73,77)(4,22,74,18)(5,118,34,114)(6,59,35,55)(7,120,36,116)(8,57,33,53)(9,21,13,17)(10,82,14,78)(11,23,15,19)(12,84,16,80)(25,67,29,61)(26,126,30,122)(27,65,31,63)(28,128,32,124)(37,49,41,45)(38,110,42,106)(39,51,43,47)(40,112,44,108)(46,98,50,102)(48,100,52,104)(54,70,58,94)(56,72,60,96)(62,86,68,90)(64,88,66,92)(69,117,93,113)(71,119,95,115)(85,125,89,121)(87,127,91,123)(97,109,101,105)(99,111,103,107)>;
G:=Group( (1,75)(2,76)(3,73)(4,74)(5,34)(6,35)(7,36)(8,33)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60)(61,67)(62,68)(63,65)(64,66)(69,93)(70,94)(71,95)(72,96)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,91,75,87)(2,88,76,92)(3,89,73,85)(4,86,74,90)(5,18,34,22)(6,23,35,19)(7,20,36,24)(8,21,33,17)(9,29,13,25)(10,26,14,30)(11,31,15,27)(12,28,16,32)(37,57,41,53)(38,54,42,58)(39,59,43,55)(40,56,44,60)(45,67,49,61)(46,62,50,68)(47,65,51,63)(48,64,52,66)(69,81,93,77)(70,78,94,82)(71,83,95,79)(72,80,96,84)(97,117,101,113)(98,114,102,118)(99,119,103,115)(100,116,104,120)(105,125,109,121)(106,122,110,126)(107,127,111,123)(108,124,112,128), (1,83,75,79)(2,24,76,20)(3,81,73,77)(4,22,74,18)(5,118,34,114)(6,59,35,55)(7,120,36,116)(8,57,33,53)(9,21,13,17)(10,82,14,78)(11,23,15,19)(12,84,16,80)(25,67,29,61)(26,126,30,122)(27,65,31,63)(28,128,32,124)(37,49,41,45)(38,110,42,106)(39,51,43,47)(40,112,44,108)(46,98,50,102)(48,100,52,104)(54,70,58,94)(56,72,60,96)(62,86,68,90)(64,88,66,92)(69,117,93,113)(71,119,95,115)(85,125,89,121)(87,127,91,123)(97,109,101,105)(99,111,103,107) );
G=PermutationGroup([[(1,75),(2,76),(3,73),(4,74),(5,34),(6,35),(7,36),(8,33),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(37,41),(38,42),(39,43),(40,44),(45,49),(46,50),(47,51),(48,52),(53,57),(54,58),(55,59),(56,60),(61,67),(62,68),(63,65),(64,66),(69,93),(70,94),(71,95),(72,96),(77,81),(78,82),(79,83),(80,84),(85,89),(86,90),(87,91),(88,92),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128)], [(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)], [(1,91,75,87),(2,88,76,92),(3,89,73,85),(4,86,74,90),(5,18,34,22),(6,23,35,19),(7,20,36,24),(8,21,33,17),(9,29,13,25),(10,26,14,30),(11,31,15,27),(12,28,16,32),(37,57,41,53),(38,54,42,58),(39,59,43,55),(40,56,44,60),(45,67,49,61),(46,62,50,68),(47,65,51,63),(48,64,52,66),(69,81,93,77),(70,78,94,82),(71,83,95,79),(72,80,96,84),(97,117,101,113),(98,114,102,118),(99,119,103,115),(100,116,104,120),(105,125,109,121),(106,122,110,126),(107,127,111,123),(108,124,112,128)], [(1,83,75,79),(2,24,76,20),(3,81,73,77),(4,22,74,18),(5,118,34,114),(6,59,35,55),(7,120,36,116),(8,57,33,53),(9,21,13,17),(10,82,14,78),(11,23,15,19),(12,84,16,80),(25,67,29,61),(26,126,30,122),(27,65,31,63),(28,128,32,124),(37,49,41,45),(38,110,42,106),(39,51,43,47),(40,112,44,108),(46,98,50,102),(48,100,52,104),(54,70,58,94),(56,72,60,96),(62,86,68,90),(64,88,66,92),(69,117,93,113),(71,119,95,115),(85,125,89,121),(87,127,91,123),(97,109,101,105),(99,111,103,107)]])
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 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | C4○D4 |
| kernel | C24.631C23 | C2×C2.C42 | C22×C4⋊C4 | C2×C4⋊C4 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 5 | 2 | 8 | 10 | 6 | 12 |
Matrix representation of C24.631C23 ►in GL7(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(7,GF(5))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1] >;
C24.631C23 in GAP, Magma, Sage, TeX
C_2^4._{631}C_2^3 % in TeX
G:=Group("C2^4.631C2^3"); // GroupNames label
G:=SmallGroup(128,173);
// by ID
G=gap.SmallGroup(128,173);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,2,448,141,64,422,387,58]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=a,a*b=b*a,a*c=c*a,a*d=d*a,f*e*f^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d>;
// generators/relations