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G = C23.669C24  order 128 = 27

386th central stem extension by C23 of C24

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

Aliases: C23.669C24, C22.3352- (1+4), C22.4422+ (1+4), C42⋊5C4.17C2, C42⋊8C4.47C2, (C2×C42).701C22, (C22×C4).587C23, C23.63C23.49C2, C23.65C23.79C2, C23.81C23.39C2, C2.C42.373C22, C23.83C23.36C2, C2.41(C22.49C24), C2.56(C22.35C24), C2.41(C22.57C24), C2.94(C22.33C24), C2.107(C22.46C24), (C2×C4).463(C4○D4), (C2×C4⋊C4).479C22, C22.530(C2×C4○D4), SmallGroup(128,1501)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.669C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C23.669C24
Lower central
C1 — C23 — C23.669C24
Upper central
C1 — C23 — C23.669C24
Jennings
C1 — C23 — C23.669C24

Subgroups: 292 in 172 conjugacy classes, 88 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×6], C2×C4 [×42], C23, C42 [×6], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×12], C2.C42 [×2], C2.C42 [×18], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C42⋊8C4, C42⋊5C4 [×2], C23.63C23 [×4], C23.65C23 [×2], C23.81C23, C23.81C23 [×2], C23.83C23, C23.83C23 [×2], C23.669C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4), 2- (1+4) [×3], C22.33C24, C22.35C24 [×2], C22.46C24 [×2], C22.49C24, C22.57C24, C23.669C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=b, f2=ba=ab, g2=a, ac=ca, ede-1=ad=da, geg-1=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, gdg-1=abd, fg=gf >

Smallest permutation representation
►Regular action on 128 points
Generators in S128
(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
003000
000300
000004
000040
,
200000
020000
002100
002300
000010
000004
,
300000
020000
004000
004100
000030
000003
,
100000
040000
004000
000400
000001
000040

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,2,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

32 conjugacy classes

class 1 2A···2G4A···4R4S···4X
order12···24···44···4
size11···14···48···8

32 irreducible representations

dim1111111244
type++++++++-
imageC1C2C2C2C2C2C2C4○D42+ (1+4)2- (1+4)
kernelC23.669C24C42⋊8C4C42⋊5C4C23.63C23C23.65C23C23.81C23C23.83C23C2×C4C22C22
# reps11242331213

In GAP, Magma, Sage, TeX 

C_2^3._{669}C_2^4
 
% in TeX
 
G:=Group("C2^3.669C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1501);
 
// by ID
 
G=gap.SmallGroup(128,1501);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,268,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=b,f^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=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,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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