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G = C429Q8order 128 = 27

9th semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C429Q8, C23.512C24, C22.2912+ 1+4, C22.2122- 1+4, C428C4.38C2, C424C4.24C2, (C2×C42).599C22, (C22×C4).850C23, C22.130(C22×Q8), (C22×Q8).150C22, C2.74(C22.45C24), C23.65C23.66C2, C2.C42.241C22, C23.78C23.14C2, C23.67C23.48C2, C23.63C23.36C2, C23.83C23.22C2, C23.81C23.25C2, C2.33(C22.49C24), C2.41(C23.37C23), C2.79(C22.46C24), C2.18(C23.41C23), (C2×C4).168(C2×Q8), (C2×C4).414(C4○D4), (C2×C4⋊C4).351C22, C22.388(C2×C4○D4), SmallGroup(128,1344)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C429Q8
C1C2C22C23C22×C4C2×C42C424C4 — C429Q8
C1C23 — C429Q8
C1C23 — C429Q8
C1C23 — C429Q8

Generators and relations for C429Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 324 in 194 conjugacy classes, 100 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×38], Q8 [×4], C23, C42 [×4], C42 [×8], C4⋊C4 [×17], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×5], C2.C42 [×2], C2.C42 [×16], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×8], C22×Q8, C424C4 [×2], C428C4, C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C23.78C23, C23.81C23 [×2], C23.83C23, C429Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C23.37C23 [×2], C23.41C23, C22.45C24, C22.46C24 [×2], C22.49C24, C429Q8

Smallest permutation representation of C429Q8
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 9 53 16)(2 10 54 13)(3 11 55 14)(4 12 56 15)(5 85 125 84)(6 86 126 81)(7 87 127 82)(8 88 128 83)(17 61 24 98)(18 62 21 99)(19 63 22 100)(20 64 23 97)(25 48 32 44)(26 45 29 41)(27 46 30 42)(28 47 31 43)(33 109 40 114)(34 110 37 115)(35 111 38 116)(36 112 39 113)(49 70 58 65)(50 71 59 66)(51 72 60 67)(52 69 57 68)(73 123 78 118)(74 124 79 119)(75 121 80 120)(76 122 77 117)(89 105 94 104)(90 106 95 101)(91 107 96 102)(92 108 93 103)
(1 49 41 98)(2 59 42 62)(3 51 43 100)(4 57 44 64)(5 102 110 122)(6 108 111 118)(7 104 112 124)(8 106 109 120)(9 70 26 17)(10 66 27 21)(11 72 28 19)(12 68 25 23)(13 71 30 18)(14 67 31 22)(15 69 32 20)(16 65 29 24)(33 80 83 90)(34 76 84 96)(35 78 81 92)(36 74 82 94)(37 77 85 91)(38 73 86 93)(39 79 87 89)(40 75 88 95)(45 61 53 58)(46 99 54 50)(47 63 55 60)(48 97 56 52)(101 114 121 128)(103 116 123 126)(105 113 119 127)(107 115 117 125)
(1 105 41 119)(2 103 42 123)(3 107 43 117)(4 101 44 121)(5 63 110 60)(6 99 111 50)(7 61 112 58)(8 97 109 52)(9 91 26 77)(10 95 27 75)(11 89 28 79)(12 93 25 73)(13 90 30 80)(14 94 31 74)(15 92 32 78)(16 96 29 76)(17 37 70 85)(18 33 71 83)(19 39 72 87)(20 35 69 81)(21 40 66 88)(22 36 67 82)(23 38 68 86)(24 34 65 84)(45 124 53 104)(46 118 54 108)(47 122 55 102)(48 120 56 106)(49 127 98 113)(51 125 100 115)(57 128 64 114)(59 126 62 116)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111112244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC429Q8C424C4C428C4C23.63C23C23.65C23C23.67C23C23.78C23C23.81C23C23.83C23C42C2×C4C22C22
# reps12142212141611

Matrix representation of C429Q8 in GL6(𝔽5)

300000
020000
001000
000100
000030
000012
,
300000
030000
004000
000400
000040
000031
,
040000
100000
003000
002200
000010
000001
,
300000
020000
003100
000200
000014
000004

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

C429Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9Q_8
% in TeX

G:=Group("C4^2:9Q8");
// GroupNames label

G:=SmallGroup(128,1344);
// by ID

G=gap.SmallGroup(128,1344);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675,248]);
// Polycyclic

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

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