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G = C42.26Q8order 128 = 27

26th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.26Q8, C81(C4⋊C4), C8⋊C49C4, C2.1(C8⋊Q8), C4.4(C4⋊Q8), (C2×C8).10Q8, (C2×C8).108D4, C2.1(C83D4), C429C4.8C2, C428C4.9C2, C42.144(C2×C4), C2.1(C8.2D4), C23.756(C2×D4), (C22×C4).280D4, C22.30(C4⋊Q8), C2.8(C429C4), C22.30(C41D4), C22.66(C8⋊C22), (C22×C8).220C22, (C2×C42).259C22, C2.9(M4(2)⋊C4), (C22×C4).1346C23, C22.55(C8.C22), C4.36(C2×C4⋊C4), (C2×C8).63(C2×C4), (C2×C8⋊C4).4C2, (C2×C4.Q8).4C2, (C2×C4).48(C4⋊C4), (C2×C4).731(C2×D4), (C2×C4).196(C2×Q8), (C2×C2.D8).33C2, (C2×C4⋊C4).50C22, C22.105(C2×C4⋊C4), (C2×C4).545(C22×C4), SmallGroup(128,579)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.26Q8
C1C2C22C23C22×C4C2×C42C2×C8⋊C4 — C42.26Q8
C1C2C2×C4 — C42.26Q8
C1C23C2×C42 — C42.26Q8
C1C2C2C22×C4 — C42.26Q8

Generators and relations for C42.26Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 236 in 128 conjugacy classes, 76 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C428C4, C429C4, C2×C8⋊C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C42.26Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C8⋊C22 [×2], C8.C22 [×2], C429C4, M4(2)⋊C4 [×2], C83D4, C8.2D4, C8⋊Q8 [×2], C42.26Q8

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

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim1111111222244
type++++++-+-++-
imageC1C2C2C2C2C2C4Q8D4Q8D4C8⋊C22C8.C22
kernelC42.26Q8C428C4C429C4C2×C8⋊C4C2×C4.Q8C2×C2.D8C8⋊C4C42C2×C8C2×C8C22×C4C22C22
# reps1111228244222

Matrix representation of C42.26Q8 in GL8(𝔽17)

41000000
013000000
00010000
001600000
00000010
0000113130
000016000
0000015154
,
10000000
01000000
00100000
00010000
000001600
00001000
000016440
000082413
,
41000000
013000000
00100000
00010000
00000001
0000915134
000002213
00000100
,
155000000
162000000
00100000
000160000
000013121316
000083144
0000112164
00007812

G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,13,0,15,0,0,0,0,1,13,0,15,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,8,0,0,0,0,16,0,4,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,15,2,1,0,0,0,0,0,13,2,0,0,0,0,0,1,4,13,0],[15,16,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,8,1,7,0,0,0,0,12,3,12,8,0,0,0,0,13,14,16,1,0,0,0,0,16,4,4,2] >;

C42.26Q8 in GAP, Magma, Sage, TeX

C_4^2._{26}Q_8
% in TeX

G:=Group("C4^2.26Q8");
// GroupNames label

G:=SmallGroup(128,579);
// by ID

G=gap.SmallGroup(128,579);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248]);
// Polycyclic

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

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