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G = C2×C23.65C23  order 128 = 27

Direct product of C2 and C23.65C23

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

Aliases: C2×C23.65C23, C23.173C24, C24.645C23, C22.30(C4×Q8), (C22×C4).96Q8, (C22×C4).562D4, C23.827(C2×D4), C22.110(C4×D4), C22.40(C4⋊Q8), C23.142(C2×Q8), (C22×C42).22C2, C22.64(C23×C4), C22.71(C22×D4), C23.361(C4○D4), C22.23(C22×Q8), (C22×C4).448C23, C23.284(C22×C4), (C23×C4).280C22, C22.88(C22⋊Q8), C22.158(C4⋊D4), (C2×C42).1088C22, C22.26(C42.C2), C2.C42.509C22, C4⋊3(C2×C4⋊C4), C2.5(C2×C4×Q8), (C2×C4⋊C4)⋊33C4, C2.10(C2×C4×D4), C4⋊C4⋊38(C2×C4), C2.2(C2×C4⋊Q8), (C2×C4)⋊11(C4⋊C4), C2.4(C2×C4⋊D4), C2.9(C22×C4⋊C4), C2.4(C2×C22⋊Q8), (C2×C4).673(C2×D4), C22.72(C2×C4⋊C4), (C2×C4).225(C2×Q8), C2.3(C2×C42.C2), (C22×C4⋊C4).22C2, C22.65(C2×C4○D4), (C2×C4⋊C4).789C22, (C22×C4).293(C2×C4), (C2×C4).207(C22×C4), (C2×C2.C42).17C2, SmallGroup(128,1023)

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

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

Subgroups: 588 in 400 conjugacy classes, 236 normal (24 characteristic)
C1, C2 [×7], C2 [×8], C4 [×8], C4 [×20], C22 [×7], C22 [×28], C2×C4 [×44], C2×C4 [×68], C23, C23 [×14], C42 [×8], C4⋊C4 [×16], C4⋊C4 [×24], C22×C4 [×42], C22×C4 [×28], C24, C2.C42 [×8], C2×C42 [×4], C2×C42 [×4], C2×C4⋊C4 [×24], C2×C4⋊C4 [×12], C23×C4 [×3], C23×C4 [×4], C2×C2.C42 [×2], C23.65C23 [×8], C22×C42, C22×C4⋊C4 [×2], C22×C4⋊C4 [×2], C2×C23.65C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], Q8 [×8], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C2×Q8 [×12], C4○D4 [×4], C24, C2×C4⋊C4 [×12], C4×D4 [×4], C4×Q8 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C23×C4, C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×2], C23.65C23 [×8], C22×C4⋊C4, C2×C4×D4, C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C42.C2, C2×C4⋊Q8, C2×C23.65C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, 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 >

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
010000
001000
000100
000010
000001
,
300000
040000
000400
004000
000001
000010
,
200000
010000
001000
000400
000002
000030
,
100000
040000
004000
000400
000004
000010

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[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],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
 

56 conjugacy classes

class 1 2A···2O4A···4X4Y···4AN
order12···24···44···4
size11···12···24···4

56 irreducible representations

dim111111222
type++++++-
imageC1C2C2C2C2C4D4Q8C4○D4
kernelC2×C23.65C23C2×C2.C42C23.65C23C22×C42C22×C4⋊C4C2×C4⋊C4C22×C4C22×C4C23
# reps1281416888

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{65}C_2^3
 
% in TeX
 
G:=Group("C2xC2^3.65C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1023);
 
// by ID
 
G=gap.SmallGroup(128,1023);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,184]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=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
 

׿
×
⋊
ℤ
𝔽
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