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G = C42.61Q8order 128 = 27

21st non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Aliases: C42.61Q8, C42.429D4, C4⋊C44C8, C42(C4⋊C8), C2.4(C8×Q8), C2.14(C8×D4), (C2×C8).39Q8, (C2×C8).227D4, C4.48(C4⋊Q8), C2.4(C86D4), C2.4(C84Q8), C22.26(C4×Q8), C22.102(C4×D4), (C2×C4).46M4(2), C4.197(C4⋊D4), C22.33(C8○D4), C4.117(C22⋊Q8), C22.43(C22×C8), (C22×C8).52C22, C4.38(C42.C2), C2.C42.24C4, (C2×C42).305C22, C23.272(C22×C4), C22.54(C2×M4(2)), (C22×C4).1635C23, C22.7C42.8C2, C2.4(C42.6C22), C2.3(C23.65C23), (C2×C4×C8).22C2, C2.12(C2×C4⋊C8), (C2×C4⋊C8).29C2, (C2×C4⋊C4).58C4, (C4×C4⋊C4).20C2, (C2×C4).22(C2×C8), C22.67(C2×C4⋊C4), (C2×C4).345(C2×Q8), (C2×C4).134(C4⋊C4), (C2×C4).1538(C2×D4), (C2×C4).941(C4○D4), (C22×C4).125(C2×C4), SmallGroup(128,671)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.61Q8
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C42.61Q8
C1C22 — C42.61Q8
C1C22×C4 — C42.61Q8
C1C2C2C22×C4 — C42.61Q8

Generators and relations for C42.61Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 130 conjugacy classes, 80 normal (38 characteristic)
C1, C2 [×7], C4 [×4], C4 [×4], C4 [×8], C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×12], C2×C4 [×12], C23, C42 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C2×C4⋊C8 [×2], C42.61Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4⋊C8 [×4], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C8○D4 [×2], C23.65C23, C2×C4⋊C8, C42.6C22, C8×D4, C86D4, C8×Q8, C84Q8, C42.61Q8

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

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim111111112222222
type++++++-+-
imageC1C2C2C2C2C4C4C8D4Q8D4Q8M4(2)C4○D4C8○D4
kernelC42.61Q8C22.7C42C4×C4⋊C4C2×C4×C8C2×C4⋊C8C2.C42C2×C4⋊C4C4⋊C4C42C42C2×C8C2×C8C2×C4C2×C4C22
# reps1211344162222448

Matrix representation of C42.61Q8 in GL5(𝔽17)

160000
00100
016000
0001615
00011
,
130000
04000
00400
00010
00001
,
90000
02000
00200
0001615
00011
,
90000
012900
09500
000139
00004

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,15,1],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,16,1,0,0,0,15,1],[9,0,0,0,0,0,12,9,0,0,0,9,5,0,0,0,0,0,13,0,0,0,0,9,4] >;

C42.61Q8 in GAP, Magma, Sage, TeX

C_4^2._{61}Q_8
% in TeX

G:=Group("C4^2.61Q8");
// GroupNames label

G:=SmallGroup(128,671);
// by ID

G=gap.SmallGroup(128,671);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,124]);
// Polycyclic

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

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