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G = C2×C42Q16order 128 = 27

Direct product of C2 and C42Q16

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

Aliases: C2×C42Q16, C42.209D4, C42.318C23, C43(C2×Q16), (C2×C4)⋊10Q16, Q8.1(C2×D4), (C2×Q8).169D4, C4.64(C22×D4), C2.5(C22×Q16), C4.67(C4⋊D4), C4⋊C8.280C22, C4⋊C4.374C23, (C2×C8).133C23, (C2×C4).237C24, C23.856(C2×D4), (C22×C4).792D4, C4⋊Q8.254C22, (C2×Q8).33C23, (C22×Q16).8C2, C22.47(C2×Q16), (C4×Q8).289C22, (C2×C42).806C22, (C22×C8).138C22, (C2×Q16).115C22, C22.497(C22×D4), C22.169(C4⋊D4), (C22×C4).1527C23, Q8⋊C4.144C22, (C22×Q8).269C22, C22.102(C8.C22), (C2×C4⋊C8).41C2, (C2×C4×Q8).48C2, (C2×C4⋊Q8).42C2, C4.147(C2×C4○D4), C2.55(C2×C4⋊D4), (C2×C4).1417(C2×D4), C2.13(C2×C8.C22), (C2×C4).904(C4○D4), (C2×C4⋊C4).918C22, (C2×Q8⋊C4).23C2, SmallGroup(128,1765)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C42Q16
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C2×C42Q16
C1C2C2×C4 — C2×C42Q16
C1C23C2×C42 — C2×C42Q16
C1C2C2C2×C4 — C2×C42Q16

Generators and relations for C2×C42Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 396 in 240 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×20], Q8 [×4], Q8 [×18], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×10], C2×Q8 [×13], Q8⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4×Q8 [×4], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16 [×8], C2×Q16 [×8], C22×Q8, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C4⋊C8, C42Q16 [×8], C2×C4×Q8, C2×C4⋊Q8, C22×Q16 [×2], C2×C42Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C42Q16 [×4], C2×C4⋊D4, C22×Q16, C2×C8.C22, C2×C42Q16

Smallest permutation representation of C2×C42Q16
Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 81)(72 82)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 121)(80 122)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 105)(96 106)
(1 82 107 73)(2 74 108 83)(3 84 109 75)(4 76 110 85)(5 86 111 77)(6 78 112 87)(7 88 105 79)(8 80 106 81)(9 26 41 101)(10 102 42 27)(11 28 43 103)(12 104 44 29)(13 30 45 97)(14 98 46 31)(15 32 47 99)(16 100 48 25)(17 51 57 118)(18 119 58 52)(19 53 59 120)(20 113 60 54)(21 55 61 114)(22 115 62 56)(23 49 63 116)(24 117 64 50)(33 66 91 125)(34 126 92 67)(35 68 93 127)(36 128 94 69)(37 70 95 121)(38 122 96 71)(39 72 89 123)(40 124 90 65)
(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 22 5 18)(2 21 6 17)(3 20 7 24)(4 19 8 23)(9 66 13 70)(10 65 14 69)(11 72 15 68)(12 71 16 67)(25 92 29 96)(26 91 30 95)(27 90 31 94)(28 89 32 93)(33 97 37 101)(34 104 38 100)(35 103 39 99)(36 102 40 98)(41 125 45 121)(42 124 46 128)(43 123 47 127)(44 122 48 126)(49 76 53 80)(50 75 54 79)(51 74 55 78)(52 73 56 77)(57 108 61 112)(58 107 62 111)(59 106 63 110)(60 105 64 109)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim1111111222224
type++++++++++--
imageC1C2C2C2C2C2C2D4D4D4Q16C4○D4C8.C22
kernelC2×C42Q16C2×Q8⋊C4C2×C4⋊C8C42Q16C2×C4×Q8C2×C4⋊Q8C22×Q16C42C22×C4C2×Q8C2×C4C2×C4C22
# reps1218112224842

Matrix representation of C2×C42Q16 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
0160000
100000
0013000
000400
000010
000001
,
100000
0160000
0001600
001000
0000150
0000118
,
100000
010000
0016000
000100
0000312
0000214

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,15,11,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,3,2,0,0,0,0,12,14] >;

C2×C42Q16 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes_2Q_{16}
% in TeX

G:=Group("C2xC4:2Q16");
// GroupNames label

G:=SmallGroup(128,1765);
// by ID

G=gap.SmallGroup(128,1765);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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