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G = C2×Q16⋊C4order 128 = 27

Direct product of C2 and Q16⋊C4

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

Aliases: C2×Q16⋊C4, C42.203D4, C42.270C23, C4.46(C4×D4), (C2×Q16)⋊17C4, Q1610(C2×C4), C4.18(C23×C4), C8.18(C22×C4), Q8.3(C22×C4), C4⋊C4.358C23, (C2×C8).409C23, (C2×C4).198C24, C22.119(C4×D4), C23.844(C2×D4), (C22×C4).708D4, (C4×Q8).273C22, (C22×Q16).15C2, (C2×Q8).341C23, C8⋊C4.110C22, C4.Q8.124C22, (C22×C8).437C22, (C2×C42).763C22, (C2×Q16).150C22, C22.142(C22×D4), (C22×C4).1514C23, Q8⋊C4.194C22, C22.97(C8.C22), (C22×Q8).462C22, C2.58(C2×C4×D4), C4.6(C2×C4○D4), (C2×C4×Q8).42C2, (C2×C8).96(C2×C4), (C2×C8⋊C4).10C2, (C2×C4.Q8).10C2, C2.5(C2×C8.C22), (C2×C4).1210(C2×D4), (C2×Q8).159(C2×C4), (C2×C4).690(C4○D4), (C2×C4⋊C4).910C22, (C2×C4).469(C22×C4), (C2×Q8⋊C4).36C2, SmallGroup(128,1673)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×Q16⋊C4
C1C2C22C2×C4C22×C4C2×C42C2×C4×Q8 — C2×Q16⋊C4
C1C2C4 — C2×Q16⋊C4
C1C23C2×C42 — C2×Q16⋊C4
C1C2C2C2×C4 — C2×Q16⋊C4

Generators and relations for C2×Q16⋊C4
 G = < a,b,c,d | a2=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, cd=dc >

Subgroups: 348 in 240 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C8⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C2×Q16, C22×Q8, C2×C8⋊C4, C2×Q8⋊C4, C2×C4.Q8, Q16⋊C4, C2×C4×Q8, C22×Q16, C2×Q16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8.C22, C23×C4, C22×D4, C2×C4○D4, Q16⋊C4, C2×C4×D4, C2×C8.C22, C2×Q16⋊C4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim111111112224
type+++++++++-
imageC1C2C2C2C2C2C2C4D4D4C4○D4C8.C22
kernelC2×Q16⋊C4C2×C8⋊C4C2×Q8⋊C4C2×C4.Q8Q16⋊C4C2×C4×Q8C22×Q16C2×Q16C42C22×C4C2×C4C22
# reps1121821162244

Matrix representation of C2×Q16⋊C4 in GL8(𝔽17)

160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
105000000
77000000
0015140000
001320000
000016400
00003100
0000301316
0000212124
,
12000000
016000000
001610000
00010000
00000010
000014001
000016000
000001630
,
130000000
013000000
001600000
000160000
00001200
0000161600
000011012
0000661616

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[10,7,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,0,15,13,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,16,3,3,2,0,0,0,0,4,1,0,12,0,0,0,0,0,0,13,12,0,0,0,0,0,0,16,4],[1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,14,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,11,6,0,0,0,0,2,16,0,6,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16] >;

C2×Q16⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xQ16:C4");
// GroupNames label

G:=SmallGroup(128,1673);
// by ID

G=gap.SmallGroup(128,1673);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,1430,184,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations

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