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G = C23.346C24  order 128 = 27

63rd central stem extension by C23 of C24

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

Aliases: C23.346C24, C22.1132- (1+4), C4⋊C4⋊12Q8, C2.12(D4×Q8), (C2×Q8).223D4, C2.6(Q8⋊3Q8), C2.21(Q8⋊5D4), C22.74(C22×Q8), (C22×C4).803C23, (C2×C42).489C22, C22.226(C22×D4), C4.79(C22.D4), (C22×Q8).103C22, C23.81C23.3C2, C23.83C23.3C2, C23.65C23.39C2, C23.63C23.13C2, C2.C42.103C22, C23.67C23.32C2, C2.15(C23.37C23), C2.12(C22.50C24), C2.10(C22.35C24), (C4×C4⋊C4).48C2, (C2×C4×Q8).29C2, (C2×C4⋊Q8).29C2, (C2×C4).29(C2×Q8), (C2×C4).327(C2×D4), (C2×C4).103(C4○D4), (C2×C4⋊C4).228C22, C22.223(C2×C4○D4), C2.24(C2×C22.D4), SmallGroup(128,1178)

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

Derived series
C1 — C23 — C23.346C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.346C24
Lower central
C1 — C23 — C23.346C24
Upper central
C1 — C23 — C23.346C24
Jennings
C1 — C23 — C23.346C24

Subgroups: 372 in 230 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×20], C22 [×7], C2×C4 [×18], C2×C4 [×36], Q8 [×12], C23, C42 [×10], C4⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×7], C22×C4 [×8], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×2], C2.C42 [×14], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C4×Q8 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C4×C4⋊C4, C23.63C23 [×4], C23.65C23, C23.67C23 [×3], C23.81C23 [×2], C23.83C23 [×2], C2×C4×Q8, C2×C4⋊Q8, C23.346C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22.D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2- (1+4) [×2], C2×C22.D4, C23.37C23, C22.35C24, Q8⋊5D4, D4×Q8, C22.50C24, Q8⋊3Q8, C23.346C24

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

Smallest permutation representation
►Regular action on 128 points
Generators in S128
(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 9)(2 10)(3 11)(4 12)(5 71)(6 72)(7 69)(8 70)(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)(41 103)(42 104)(43 101)(44 102)(45 107)(46 108)(47 105)(48 106)(49 111)(50 112)(51 109)(52 110)(53 115)(54 116)(55 113)(56 114)(57 119)(58 120)(59 117)(60 118)(61 123)(62 124)(63 121)(64 122)(65 126)(66 127)(67 128)(68 125)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
430000
004300
000100
000010
000001
,
140000
040000
003000
000300
000040
000001
,
400000
040000
003000
002200
000001
000010
,
320000
020000
001000
000100
000010
000001

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim1111111112224
type+++++++++-+-
imageC1C2C2C2C2C2C2C2C2Q8D4C4○D42- (1+4)
kernelC23.346C24C4×C4⋊C4C23.63C23C23.65C23C23.67C23C23.81C23C23.83C23C2×C4×Q8C2×C4⋊Q8C4⋊C4C2×Q8C2×C4C22
# reps11413221144122

In GAP, Magma, Sage, TeX 

C_2^3._{346}C_2^4
 
% in TeX
 
G:=Group("C2^3.346C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1178);
 
// by ID
 
G=gap.SmallGroup(128,1178);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=a,e^2=f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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