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G = C42.59Q8order 128 = 27

19th non-split extension by C42 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.59Q8, C84(C4⋊C4), (C4×C8)⋊17C4, (C2×C4).74D8, C42(C2.D8), C4.2(C4⋊Q8), (C2×C8).42Q8, (C2×C8).243D4, (C2×C4).36Q16, C2.1(C84D4), C2.1(C82Q8), C22.35(C2×D8), C42.324(C2×C4), C429C4.7C2, C2.1(C4⋊Q16), C23.754(C2×D4), (C22×C4).578D4, C22.28(C4⋊Q8), C22.28(C2×Q16), C2.6(C429C4), C22.28(C41D4), (C22×C8).524C22, (C2×C42).1060C22, (C22×C4).1344C23, (C2×C4×C8).34C2, C4.34(C2×C4⋊C4), C2.8(C2×C2.D8), (C2×C2.D8).2C2, (C2×C8).221(C2×C4), (C2×C4).729(C2×D4), (C2×C4).194(C2×Q8), (C2×C4).132(C4⋊C4), (C2×C4⋊C4).48C22, C22.103(C2×C4⋊C4), (C2×C4).543(C22×C4), SmallGroup(128,577)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.59Q8
C1C2C22C23C22×C4C2×C42C2×C4×C8 — C42.59Q8
C1C2C2×C4 — C42.59Q8
C1C23C2×C42 — C42.59Q8
C1C2C2C22×C4 — C42.59Q8

Generators and relations for C42.59Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=bc2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 252 in 140 conjugacy classes, 92 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×4], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C4×C8 [×4], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C22×C8 [×2], C429C4 [×2], C2×C4×C8, C2×C2.D8 [×4], C42.59Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], D8 [×4], Q16 [×4], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2.D8 [×8], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C2×D8 [×2], C2×Q16 [×2], C429C4, C2×C2.D8 [×2], C84D4, C4⋊Q16, C82Q8 [×2], C42.59Q8

Smallest permutation representation of C42.59Q8
Regular action on 128 points
Generators in S128
(1 25 56 24)(2 26 49 17)(3 27 50 18)(4 28 51 19)(5 29 52 20)(6 30 53 21)(7 31 54 22)(8 32 55 23)(9 41 110 94)(10 42 111 95)(11 43 112 96)(12 44 105 89)(13 45 106 90)(14 46 107 91)(15 47 108 92)(16 48 109 93)(33 82 115 128)(34 83 116 121)(35 84 117 122)(36 85 118 123)(37 86 119 124)(38 87 120 125)(39 88 113 126)(40 81 114 127)(57 71 103 73)(58 72 104 74)(59 65 97 75)(60 66 98 76)(61 67 99 77)(62 68 100 78)(63 69 101 79)(64 70 102 80)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 128 13 124)(10 121 14 125)(11 122 15 126)(12 123 16 127)(17 76 21 80)(18 77 22 73)(19 78 23 74)(20 79 24 75)(25 65 29 69)(26 66 30 70)(27 67 31 71)(28 68 32 72)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)(49 98 53 102)(50 99 54 103)(51 100 55 104)(52 101 56 97)(81 105 85 109)(82 106 86 110)(83 107 87 111)(84 108 88 112)(89 118 93 114)(90 119 94 115)(91 120 95 116)(92 113 96 117)
(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)
(1 107 61 81)(2 106 62 88)(3 105 63 87)(4 112 64 86)(5 111 57 85)(6 110 58 84)(7 109 59 83)(8 108 60 82)(9 104 122 53)(10 103 123 52)(11 102 124 51)(12 101 125 50)(13 100 126 49)(14 99 127 56)(15 98 128 55)(16 97 121 54)(17 90 78 113)(18 89 79 120)(19 96 80 119)(20 95 73 118)(21 94 74 117)(22 93 75 116)(23 92 76 115)(24 91 77 114)(25 46 67 40)(26 45 68 39)(27 44 69 38)(28 43 70 37)(29 42 71 36)(30 41 72 35)(31 48 65 34)(32 47 66 33)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim11111222222
type++++-+-++-
imageC1C2C2C2C4Q8D4Q8D4D8Q16
kernelC42.59Q8C429C4C2×C4×C8C2×C2.D8C4×C8C42C2×C8C2×C8C22×C4C2×C4C2×C4
# reps12148244288

Matrix representation of C42.59Q8 in GL5(𝔽17)

160000
00100
016000
00010
00001
,
160000
016000
001600
00012
0001616
,
10000
001600
01000
00066
000140
,
130000
061300
0131100
0001010
000127

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,6,14,0,0,0,6,0],[13,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,10,12,0,0,0,10,7] >;

C42.59Q8 in GAP, Magma, Sage, TeX

C_4^2._{59}Q_8
% in TeX

G:=Group("C4^2.59Q8");
// GroupNames label

G:=SmallGroup(128,577);
// by ID

G=gap.SmallGroup(128,577);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,100,2019,248]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;
// generators/relations

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