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G = C23.409C24  order 128 = 27

126th central stem extension by C23 of C24

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

Aliases: C23.409C24, C22.2042+ (1+4), C22.1552- (1+4), C4⋊C4.22Q8, C42⋊8C4.27C2, C2.12(Q8⋊3Q8), C2.25(D4⋊3Q8), C4.31(C42.C2), C22.92(C22×Q8), (C22×C4).831C23, (C2×C42).529C22, C23.83C23.7C2, C23.63C23.20C2, C2.C42.160C22, C23.65C23.46C2, C2.33(C22.36C24), C2.25(C22.50C24), C2.39(C22.47C24), C2.54(C23.36C23), (C4×C4⋊C4).55C2, (C2×C4).45(C2×Q8), C2.14(C2×C42.C2), (C2×C4).131(C4○D4), (C2×C4⋊C4).275C22, C22.286(C2×C4○D4), SmallGroup(128,1241)

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

Derived series
C1 — C23 — C23.409C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.409C24
Lower central
C1 — C23 — C23.409C24
Upper central
C1 — C23 — C23.409C24
Jennings
C1 — C23 — C23.409C24

Subgroups: 308 in 190 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×18], C22 [×7], C2×C4 [×14], C2×C4 [×38], C23, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×7], C22×C4 [×8], C2.C42 [×2], C2.C42 [×14], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×6], C2×C4⋊C4 [×8], C4×C4⋊C4 [×2], C42⋊8C4, C23.63C23 [×2], C23.65C23 [×4], C23.65C23 [×2], C23.83C23 [×4], C23.409C24

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C2×C42.C2, C23.36C23, C22.36C24, C22.47C24, D4⋊3Q8, C22.50C24, Q8⋊3Q8, C23.409C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=b, f2=ba=ab, g2=a, 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 77)(2 78)(3 79)(4 80)(5 127)(6 128)(7 125)(8 126)(9 39)(10 40)(11 37)(12 38)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)(29 36)(30 33)(31 34)(32 35)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(65 70)(66 71)(67 72)(68 69)(73 97)(74 98)(75 99)(76 100)(81 86)(82 87)(83 88)(84 85)(89 94)(90 95)(91 96)(92 93)(101 108)(102 105)(103 106)(104 107)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)

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

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
030000
300000
003200
001200
000021
000003
,
020000
200000
002000
000200
000034
000032
,
040000
100000
002000
004300
000020
000002
,
400000
040000
004000
000400
000042
000041

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

38 conjugacy classes

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

38 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ (1+4)2- (1+4)
kernelC23.409C24C4×C4⋊C4C42⋊8C4C23.63C23C23.65C23C23.83C23C4⋊C4C2×C4C22C22
# reps12126441611

In GAP, Magma, Sage, TeX 

C_2^3._{409}C_2^4
 
% in TeX
 
G:=Group("C2^3.409C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1241);
 
// by ID
 
G=gap.SmallGroup(128,1241);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,268,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=b,f^2=b*a=a*b,g^2=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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