Copied to
clipboard

G = Q84C42order 128 = 27

The semidirect product of Q8 and C42 acting through Inn(Q8)

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

Aliases: Q84C42, C23.158C24, C22.172- 1+4, C22.292+ 1+4, (C4×Q8)⋊16C4, C4.11(C2×C42), C42.174(C2×C4), Q8(C2.C42), C424C4.10C2, C2.11(C22×C42), C22.30(C23×C4), (C2×C42).401C22, (C22×C4).1234C23, (C22×Q8).506C22, C2.C42.569C22, C2.3(C23.33C23), C2.2(C23.32C23), (C4×C4⋊C4).30C2, (C2×C4×Q8).18C2, C4⋊C4.237(C2×C4), (C2×Q8).217(C2×C4), (C2×C4⋊C4).972C22, C4⋊C42(C2.C42), (C2×C4).482(C22×C4), SmallGroup(128,1008)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — Q84C42
C1C2C22C23C22×C4C2×C42C424C4 — Q84C42
C1C2 — Q84C42
C1C23 — Q84C42
C1C23 — Q84C42

Generators and relations for Q84C42
 G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, bd=db, cd=dc >

Subgroups: 380 in 314 conjugacy classes, 260 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×24], C22, C22 [×6], C2×C4 [×42], C2×C4 [×24], Q8 [×16], C23, C42 [×36], C42 [×6], C4⋊C4 [×36], C22×C4 [×15], C2×Q8 [×12], C2.C42, C2.C42 [×9], C2×C42 [×15], C2×C4⋊C4 [×9], C4×Q8 [×24], C22×Q8, C424C4 [×3], C4×C4⋊C4 [×9], C2×C4×Q8 [×3], Q84C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], 2+ 1+4, 2- 1+4 [×3], C22×C42, C23.32C23 [×3], C23.33C23 [×3], Q84C42

Smallest permutation representation of Q84C42
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)
(1 101 3 103)(2 104 4 102)(5 45 7 47)(6 48 8 46)(9 55 11 53)(10 54 12 56)(13 49 15 51)(14 52 16 50)(17 96 19 94)(18 95 20 93)(21 108 23 106)(22 107 24 105)(25 98 27 100)(26 97 28 99)(29 81 31 83)(30 84 32 82)(33 89 35 91)(34 92 36 90)(37 79 39 77)(38 78 40 80)(41 87 43 85)(42 86 44 88)(57 128 59 126)(58 127 60 125)(61 115 63 113)(62 114 64 116)(65 109 67 111)(66 112 68 110)(69 123 71 121)(70 122 72 124)(73 117 75 119)(74 120 76 118)
(1 45 22 55)(2 46 23 56)(3 47 24 53)(4 48 21 54)(5 107 9 101)(6 108 10 102)(7 105 11 103)(8 106 12 104)(13 97 126 93)(14 98 127 94)(15 99 128 95)(16 100 125 96)(17 50 27 58)(18 51 28 59)(19 52 25 60)(20 49 26 57)(29 66 39 64)(30 67 40 61)(31 68 37 62)(32 65 38 63)(33 76 41 72)(34 73 42 69)(35 74 43 70)(36 75 44 71)(77 114 81 110)(78 115 82 111)(79 116 83 112)(80 113 84 109)(85 124 91 118)(86 121 92 119)(87 122 89 120)(88 123 90 117)
(1 87 19 80)(2 86 20 79)(3 85 17 78)(4 88 18 77)(5 70 14 63)(6 69 15 62)(7 72 16 61)(8 71 13 64)(9 74 127 65)(10 73 128 68)(11 76 125 67)(12 75 126 66)(21 90 28 81)(22 89 25 84)(23 92 26 83)(24 91 27 82)(29 106 36 97)(30 105 33 100)(31 108 34 99)(32 107 35 98)(37 102 42 95)(38 101 43 94)(39 104 44 93)(40 103 41 96)(45 122 52 113)(46 121 49 116)(47 124 50 115)(48 123 51 114)(53 118 58 111)(54 117 59 110)(55 120 60 109)(56 119 57 112)

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

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

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

68 conjugacy classes

class 1 2A···2G4A···4BH
order12···24···4
size11···12···2

68 irreducible representations

dim1111144
type+++++-
imageC1C2C2C2C42+ 1+42- 1+4
kernelQ84C42C424C4C4×C4⋊C4C2×C4×Q8C4×Q8C22C22
# reps13934813

Matrix representation of Q84C42 in GL6(𝔽5)

400000
040000
001300
001400
004320
003243
,
400000
040000
001111
000412
001202
002330
,
300000
040000
003324
001411
004204
004043
,
200000
020000
004121
000002
004212
000200

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

Q84C42 in GAP, Magma, Sage, TeX

Q_8\rtimes_4C_4^2
% in TeX

G:=Group("Q8:4C4^2");
// GroupNames label

G:=SmallGroup(128,1008);
// by ID

G=gap.SmallGroup(128,1008);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,268,675,80]);
// Polycyclic

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

׿
×
𝔽