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G = Q8⋊C42order 128 = 27

3rd semidirect product of Q8 and C42 acting via C42/C2×C4=C2

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

Aliases: Q83C42, C42.94D4, (C4×Q8)⋊12C4, Q8⋊C49C4, C4.111(C4×D4), C4.6(C2×C42), C22.89(C4×D4), C42.129(C2×C4), C23.733(C2×D4), (C22×C4).672D4, C4.6(C42⋊C2), C2.2(Q16⋊C4), C2.3(SD16⋊C4), C22.55(C8⋊C22), C22.4Q16.45C2, (C22×C8).381C22, (C2×C42).236C22, (C22×C4).1310C23, C2.3(C23.38D4), C2.4(C23.36D4), C22.44(C8.C22), (C22×Q8).377C22, (C4×C4⋊C4).6C2, (C2×C4×Q8).9C2, C4⋊C4.141(C2×C4), (C2×C8).133(C2×C4), C2.21(C4×C22⋊C4), (C2×C8⋊C4).23C2, (C2×C4).1305(C2×D4), (C2×Q8).182(C2×C4), (C2×C4).541(C4○D4), (C2×C4⋊C4).749C22, (C2×C4).353(C22×C4), (C2×Q8⋊C4).31C2, (C2×C4).328(C22⋊C4), C22.122(C2×C22⋊C4), SmallGroup(128,495)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q8⋊C42
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — Q8⋊C42
C1C2C4 — Q8⋊C42
C1C23C2×C42 — Q8⋊C42
C1C2C2C22×C4 — Q8⋊C42

Generators and relations for Q8⋊C42
 G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, cd=dc >

Subgroups: 268 in 162 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×14], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×3], C2.C42, C8⋊C4 [×2], Q8⋊C4 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×2], C4×C4⋊C4, C2×C8⋊C4, C2×Q8⋊C4 [×2], C2×C4×Q8, Q8⋊C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C8⋊C22, C8.C22 [×3], C4×C22⋊C4, C23.36D4, C23.38D4, SD16⋊C4 [×2], Q16⋊C4 [×2], Q8⋊C42

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

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4AB8A···8H
order12···24···44···48···8
size11···12···24···44···4

44 irreducible representations

dim1111111122244
type+++++++++-
imageC1C2C2C2C2C2C4C4D4D4C4○D4C8⋊C22C8.C22
kernelQ8⋊C42C22.4Q16C4×C4⋊C4C2×C8⋊C4C2×Q8⋊C4C2×C4×Q8Q8⋊C4C4×Q8C42C22×C4C2×C4C22C22
# reps12112116822413

Matrix representation of Q8⋊C42 in GL8(𝔽17)

160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
1015000000
77000000
0011160000
00160000
00007100
000011000
000000167
00000071
,
85000000
49000000
00160000
0011160000
000000110
0000001016
000016700
00007100
,
40000000
04000000
001600000
000160000
00000100
000016000
000000016
00000010

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[10,7,0,0,0,0,0,0,15,7,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,16,6,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1],[8,4,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,6,16,0,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0] >;

Q8⋊C42 in GAP, Magma, Sage, TeX

Q_8\rtimes C_4^2
% in TeX

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

G:=SmallGroup(128,495);
// by ID

G=gap.SmallGroup(128,495);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,436,2019,248,4037,124]);
// Polycyclic

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

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