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G = C2×C4⋊C16order 128 = 27

Direct product of C2 and C4⋊C16

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

Aliases: C2×C4⋊C16, C42.11C8, C22.10M5(2), C8(C4⋊C16), C4(C4⋊C16), (C2×C4)⋊3C16, C42(C2×C16), (C4×C8).22C4, C4.28(C4⋊C8), C8.45(C4⋊C4), (C2×C8).66Q8, C8.40(C2×Q8), (C2×C8).403D4, C8.133(C2×D4), C23.45(C2×C8), (C2×C42).47C4, C2.2(C22×C16), (C22×C16).8C2, (C22×C8).30C4, (C22×C4).14C8, C2.4(C2×M5(2)), C22.23(C4⋊C8), C22.10(C2×C16), (C2×C8).626C23, (C4×C8).372C22, C42.327(C2×C4), (C2×C16).64C22, (C2×C4).93M4(2), C4.64(C2×M4(2)), C22.27(C22×C8), (C22×C8).593C22, C2.3(C2×C4⋊C8), (C2×C4×C8).27C2, (C2×C4)(C4⋊C16), (C2×C8)(C4⋊C16), C4.78(C2×C4⋊C4), (C2×C4).87(C2×C8), (C2×C8).215(C2×C4), (C2×C4).167(C4⋊C4), (C2×C4).611(C22×C4), (C22×C4).506(C2×C4), SmallGroup(128,881)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C4⋊C16
C1C2C4C8C2×C8C22×C8C2×C4×C8 — C2×C4⋊C16
C1C2 — C2×C4⋊C16
C1C22×C8 — C2×C4⋊C16
C1C2C2C2C2C4C4C2×C8 — C2×C4⋊C16

Generators and relations for C2×C4⋊C16
 G = < a,b,c | a2=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 108 in 92 conjugacy classes, 76 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×2], C23, C16 [×4], C42 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4 [×3], C4×C8 [×4], C2×C16 [×4], C2×C16 [×4], C2×C42, C22×C8 [×2], C4⋊C16 [×4], C2×C4×C8, C22×C16 [×2], C2×C4⋊C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C16 [×4], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C16 [×6], M5(2) [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C4⋊C16 [×4], C2×C4⋊C8, C22×C16, C2×M5(2), C2×C4⋊C16

Smallest permutation representation of C2×C4⋊C16
Regular action on 128 points
Generators in S128
(1 108)(2 109)(3 110)(4 111)(5 112)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 63)(18 64)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
(1 25 67 95)(2 96 68 26)(3 27 69 81)(4 82 70 28)(5 29 71 83)(6 84 72 30)(7 31 73 85)(8 86 74 32)(9 17 75 87)(10 88 76 18)(11 19 77 89)(12 90 78 20)(13 21 79 91)(14 92 80 22)(15 23 65 93)(16 94 66 24)(33 102 49 115)(34 116 50 103)(35 104 51 117)(36 118 52 105)(37 106 53 119)(38 120 54 107)(39 108 55 121)(40 122 56 109)(41 110 57 123)(42 124 58 111)(43 112 59 125)(44 126 60 97)(45 98 61 127)(46 128 62 99)(47 100 63 113)(48 114 64 101)
(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)

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

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

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

80 conjugacy classes

class 1 2A···2G4A···4H4I···4P8A···8P8Q···8X16A···16AF
order12···24···44···48···88···816···16
size11···11···12···21···12···22···2

80 irreducible representations

dim11111111112222
type+++++-
imageC1C2C2C2C4C4C4C8C8C16D4Q8M4(2)M5(2)
kernelC2×C4⋊C16C4⋊C16C2×C4×C8C22×C16C4×C8C2×C42C22×C8C42C22×C4C2×C4C2×C8C2×C8C2×C4C22
# reps141242288322248

Matrix representation of C2×C4⋊C16 in GL4(𝔽17) generated by

1000
01600
0010
0001
,
16000
01600
0040
00013
,
14000
01300
0001
0090
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[14,0,0,0,0,13,0,0,0,0,0,9,0,0,1,0] >;

C2×C4⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4:C16");
// GroupNames label

G:=SmallGroup(128,881);
// by ID

G=gap.SmallGroup(128,881);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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