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G = C42.39Q8order 128 = 27

39th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.39Q8, C23.547C24, C22.2392- 1+4, C22.3222+ 1+4, C4.8(C42.C2), C428C4.39C2, C424C4.27C2, C429C4.35C2, (C2×C42).621C22, (C22×C4).157C23, C22.136(C22×Q8), C2.C42.266C22, C23.65C23.68C2, C23.83C23.26C2, C23.81C23.27C2, C2.23(C23.41C23), C2.55(C22.36C24), C2.31(C22.35C24), C2.31(C22.34C24), (C2×C4).133(C2×Q8), C2.20(C2×C42.C2), (C2×C4).667(C4○D4), (C2×C4⋊C4).373C22, C22.419(C2×C4○D4), SmallGroup(128,1379)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.39Q8
C1C2C22C23C22×C4C2×C4⋊C4C23.65C23 — C42.39Q8
C1C23 — C42.39Q8
C1C23 — C42.39Q8
C1C23 — C42.39Q8

Generators and relations for C42.39Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 308 in 180 conjugacy classes, 100 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C2×C4 [×10], C2×C4 [×40], C23, C42 [×4], C42 [×4], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×12], C2.C42 [×16], C2×C42 [×3], C2×C4⋊C4 [×16], C424C4, C428C4, C429C4, C23.65C23 [×4], C23.81C23 [×4], C23.83C23 [×4], C42.39Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C2×C42.C2, C22.34C24, C22.35C24, C22.36C24 [×2], C23.41C23 [×2], C42.39Q8

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

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4P4Q···4X
order12···244444···44···4
size11···122224···48···8

32 irreducible representations

dim11111112244
type+++++++-+-
imageC1C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC42.39Q8C424C4C428C4C429C4C23.65C23C23.81C23C23.83C23C42C2×C4C22C22
# reps11114444822

Matrix representation of C42.39Q8 in GL8(𝔽5)

44000000
21000000
00400000
00040000
00004220
00000124
00004041
00003201
,
10000000
01000000
00400000
00040000
00003000
00002200
00003130
00000332
,
20000000
13000000
00320000
00020000
00003100
00000200
00000242
00000441
,
22000000
13000000
00140000
00240000
00002110
00003133
00002202
00000332

G:=sub<GL(8,GF(5))| [4,2,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,3,0,0,0,0,2,1,0,2,0,0,0,0,2,2,4,0,0,0,0,0,0,4,1,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,2,3,0,0,0,0,0,0,2,1,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2],[2,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,2,4,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1],[2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,3,2,0,0,0,0,0,1,1,2,3,0,0,0,0,1,3,0,3,0,0,0,0,0,3,2,2] >;

C42.39Q8 in GAP, Magma, Sage, TeX

C_4^2._{39}Q_8
% in TeX

G:=Group("C4^2.39Q8");
// GroupNames label

G:=SmallGroup(128,1379);
// by ID

G=gap.SmallGroup(128,1379);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,184,185,80]);
// Polycyclic

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

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