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G = C2×C4.Q16order 128 = 27

Direct product of C2 and C4.Q16

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

Aliases: C2×C4.Q16, C42.217D4, C42.330C23, Q82(C2×Q8), (C2×Q8)⋊14Q8, (C2×C4).64Q16, C4.45(C2×Q16), C4⋊C4.37C23, C2.7(C22×Q16), C4.25(C22×Q8), C4⋊C8.282C22, (C2×C4).272C24, (C2×C8).137C23, (C22×C4).798D4, C23.868(C2×D4), C4⋊Q8.259C22, C4.65(C22⋊Q8), C22.48(C2×Q16), (C2×Q8).362C23, (C4×Q8).294C22, C2.D8.164C22, (C22×C8).143C22, (C2×C42).818C22, C22.532(C22×D4), C22.119(C8⋊C22), (C22×C4).1542C23, Q8⋊C4.147C22, C22.100(C22⋊Q8), (C22×Q8).472C22, (C2×C4⋊C8).42C2, (C2×C4×Q8).50C2, C4.82(C2×C4○D4), (C2×C4⋊Q8).44C2, C2.22(C2×C8⋊C22), (C2×C4).320(C2×Q8), (C2×C2.D8).27C2, C2.53(C2×C22⋊Q8), (C2×C4).1434(C2×D4), (C2×C4).838(C4○D4), (C2×C4⋊C4).601C22, (C2×Q8⋊C4).24C2, SmallGroup(128,1806)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C4.Q16
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C2×C4.Q16
C1C2C2×C4 — C2×C4.Q16
C1C23C2×C42 — C2×C4.Q16
C1C2C2C2×C4 — C2×C4.Q16

Generators and relations for C2×C4.Q16
 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=b2c-1 >

Subgroups: 348 in 208 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 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×11], Q8⋊C4 [×8], C4⋊C8 [×4], C2.D8 [×8], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22×Q8, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C2.D8 [×2], C4.Q16 [×8], C2×C4×Q8, C2×C4⋊Q8, C2×C4.Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C2×Q16 [×6], C8⋊C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C4.Q16 [×4], C2×C22⋊Q8, C22×Q16, C2×C8⋊C22, C2×C4.Q16

Smallest permutation representation of C2×C4.Q16
Regular action on 128 points
Generators in S128
(1 93)(2 94)(3 95)(4 96)(5 89)(6 90)(7 91)(8 92)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(65 120)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 97)(80 98)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)
(1 105 11 86)(2 87 12 106)(3 107 13 88)(4 81 14 108)(5 109 15 82)(6 83 16 110)(7 111 9 84)(8 85 10 112)(17 36 128 94)(18 95 121 37)(19 38 122 96)(20 89 123 39)(21 40 124 90)(22 91 125 33)(23 34 126 92)(24 93 127 35)(25 65 100 48)(26 41 101 66)(27 67 102 42)(28 43 103 68)(29 69 104 44)(30 45 97 70)(31 71 98 46)(32 47 99 72)(49 63 116 78)(50 79 117 64)(51 57 118 80)(52 73 119 58)(53 59 120 74)(54 75 113 60)(55 61 114 76)(56 77 115 62)
(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 62 5 58)(2 76 6 80)(3 60 7 64)(4 74 8 78)(9 79 13 75)(10 63 14 59)(11 77 15 73)(12 61 16 57)(17 42 21 46)(18 66 22 70)(19 48 23 44)(20 72 24 68)(25 34 29 38)(26 91 30 95)(27 40 31 36)(28 89 32 93)(33 97 37 101)(35 103 39 99)(41 125 45 121)(43 123 47 127)(49 81 53 85)(50 107 54 111)(51 87 55 83)(52 105 56 109)(65 126 69 122)(67 124 71 128)(82 119 86 115)(84 117 88 113)(90 98 94 102)(92 104 96 100)(106 114 110 118)(108 120 112 116)

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

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

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

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+++++++++--+
imageC1C2C2C2C2C2C2D4D4Q8Q16C4○D4C8⋊C22
kernelC2×C4.Q16C2×Q8⋊C4C2×C4⋊C8C2×C2.D8C4.Q16C2×C4×Q8C2×C4⋊Q8C42C22×C4C2×Q8C2×C4C2×C4C22
# reps1212811224842

Matrix representation of C2×C4.Q16 in GL5(𝔽17)

160000
016000
001600
00010
00001
,
10000
013000
00400
00010
00001
,
10000
001600
016000
0001111
00030
,
10000
016000
00100
000615
0001011

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,11,3,0,0,0,11,0],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,6,10,0,0,0,15,11] >;

C2×C4.Q16 in GAP, Magma, Sage, TeX

C_2\times C_4.Q_{16}
% in TeX

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

G:=SmallGroup(128,1806);
// by ID

G=gap.SmallGroup(128,1806);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,4037,1027,124]);
// 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=b^2*c^-1>;
// generators/relations

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