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G = C4×C4⋊C8order 128 = 27

Direct product of C4 and C4⋊C8

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

Aliases: C4×C4⋊C8, C427C8, C43.6C2, C42.67Q8, C42.459D4, C41(C4×C8), C4.41(C4×Q8), C4.165(C4×D4), (C2×C4).71C42, (C2×C42).34C4, C2.4(C4×M4(2)), C42.263(C2×C4), (C2×C4).91M4(2), C22.21(C22×C8), C22.29(C2×C42), (C22×C8).475C22, (C2×C42).992C22, C23.256(C22×C4), C22.40(C2×M4(2)), C2.3(C42.12C4), C44(C22.7C42), (C22×C4).1609C23, C22.51(C42⋊C2), C22.7C42.48C2, C423(C22.7C42), C2.8(C2×C4×C8), C2.2(C2×C4⋊C8), C2.2(C4×C4⋊C4), C42(C2×C4⋊C8), (C2×C4×C8).14C2, (C2×C4⋊C8).59C2, (C2×C4).83(C2×C8), (C2×C8).158(C2×C4), C22.55(C2×C4⋊C4), (C2×C4).329(C2×Q8), (C2×C4).163(C4⋊C4), (C2×C4).1501(C2×D4), (C2×C4).919(C4○D4), (C22×C4).438(C2×C4), (C2×C4).599(C22×C4), SmallGroup(128,498)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C4⋊C8
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C4×C4⋊C8
C1C2 — C4×C4⋊C8
C1C2×C42 — C4×C4⋊C8
C1C2C2C22×C4 — C4×C4⋊C8

Generators and relations for C4×C4⋊C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 196 in 158 conjugacy classes, 120 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C4 [×6], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×28], C2×C4 [×6], C23, C42 [×16], C42 [×6], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×4], C4×C8 [×4], C4⋊C8 [×8], C2×C42 [×3], C2×C42 [×4], C22×C8 [×4], C22.7C42 [×2], C43, C2×C4×C8 [×2], C2×C4⋊C8 [×2], C4×C4⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C2×C8 [×12], M4(2) [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C4×C8 [×4], C4⋊C8 [×8], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C4×C4⋊C4, C2×C4×C8, C4×M4(2), C2×C4⋊C8 [×2], C42.12C4 [×2], C4×C4⋊C8

Smallest permutation representation of C4×C4⋊C8
Regular action on 128 points
Generators in S128
(1 33 95 85)(2 34 96 86)(3 35 89 87)(4 36 90 88)(5 37 91 81)(6 38 92 82)(7 39 93 83)(8 40 94 84)(9 122 75 67)(10 123 76 68)(11 124 77 69)(12 125 78 70)(13 126 79 71)(14 127 80 72)(15 128 73 65)(16 121 74 66)(17 98 41 29)(18 99 42 30)(19 100 43 31)(20 101 44 32)(21 102 45 25)(22 103 46 26)(23 104 47 27)(24 97 48 28)(49 106 117 59)(50 107 118 60)(51 108 119 61)(52 109 120 62)(53 110 113 63)(54 111 114 64)(55 112 115 57)(56 105 116 58)
(1 117 25 70)(2 71 26 118)(3 119 27 72)(4 65 28 120)(5 113 29 66)(6 67 30 114)(7 115 31 68)(8 69 32 116)(9 18 64 38)(10 39 57 19)(11 20 58 40)(12 33 59 21)(13 22 60 34)(14 35 61 23)(15 24 62 36)(16 37 63 17)(41 74 81 110)(42 111 82 75)(43 76 83 112)(44 105 84 77)(45 78 85 106)(46 107 86 79)(47 80 87 108)(48 109 88 73)(49 102 125 95)(50 96 126 103)(51 104 127 89)(52 90 128 97)(53 98 121 91)(54 92 122 99)(55 100 123 93)(56 94 124 101)
(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)| (1,33,95,85)(2,34,96,86)(3,35,89,87)(4,36,90,88)(5,37,91,81)(6,38,92,82)(7,39,93,83)(8,40,94,84)(9,122,75,67)(10,123,76,68)(11,124,77,69)(12,125,78,70)(13,126,79,71)(14,127,80,72)(15,128,73,65)(16,121,74,66)(17,98,41,29)(18,99,42,30)(19,100,43,31)(20,101,44,32)(21,102,45,25)(22,103,46,26)(23,104,47,27)(24,97,48,28)(49,106,117,59)(50,107,118,60)(51,108,119,61)(52,109,120,62)(53,110,113,63)(54,111,114,64)(55,112,115,57)(56,105,116,58), (1,117,25,70)(2,71,26,118)(3,119,27,72)(4,65,28,120)(5,113,29,66)(6,67,30,114)(7,115,31,68)(8,69,32,116)(9,18,64,38)(10,39,57,19)(11,20,58,40)(12,33,59,21)(13,22,60,34)(14,35,61,23)(15,24,62,36)(16,37,63,17)(41,74,81,110)(42,111,82,75)(43,76,83,112)(44,105,84,77)(45,78,85,106)(46,107,86,79)(47,80,87,108)(48,109,88,73)(49,102,125,95)(50,96,126,103)(51,104,127,89)(52,90,128,97)(53,98,121,91)(54,92,122,99)(55,100,123,93)(56,94,124,101), (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( (1,33,95,85)(2,34,96,86)(3,35,89,87)(4,36,90,88)(5,37,91,81)(6,38,92,82)(7,39,93,83)(8,40,94,84)(9,122,75,67)(10,123,76,68)(11,124,77,69)(12,125,78,70)(13,126,79,71)(14,127,80,72)(15,128,73,65)(16,121,74,66)(17,98,41,29)(18,99,42,30)(19,100,43,31)(20,101,44,32)(21,102,45,25)(22,103,46,26)(23,104,47,27)(24,97,48,28)(49,106,117,59)(50,107,118,60)(51,108,119,61)(52,109,120,62)(53,110,113,63)(54,111,114,64)(55,112,115,57)(56,105,116,58), (1,117,25,70)(2,71,26,118)(3,119,27,72)(4,65,28,120)(5,113,29,66)(6,67,30,114)(7,115,31,68)(8,69,32,116)(9,18,64,38)(10,39,57,19)(11,20,58,40)(12,33,59,21)(13,22,60,34)(14,35,61,23)(15,24,62,36)(16,37,63,17)(41,74,81,110)(42,111,82,75)(43,76,83,112)(44,105,84,77)(45,78,85,106)(46,107,86,79)(47,80,87,108)(48,109,88,73)(49,102,125,95)(50,96,126,103)(51,104,127,89)(52,90,128,97)(53,98,121,91)(54,92,122,99)(55,100,123,93)(56,94,124,101), (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([(1,33,95,85),(2,34,96,86),(3,35,89,87),(4,36,90,88),(5,37,91,81),(6,38,92,82),(7,39,93,83),(8,40,94,84),(9,122,75,67),(10,123,76,68),(11,124,77,69),(12,125,78,70),(13,126,79,71),(14,127,80,72),(15,128,73,65),(16,121,74,66),(17,98,41,29),(18,99,42,30),(19,100,43,31),(20,101,44,32),(21,102,45,25),(22,103,46,26),(23,104,47,27),(24,97,48,28),(49,106,117,59),(50,107,118,60),(51,108,119,61),(52,109,120,62),(53,110,113,63),(54,111,114,64),(55,112,115,57),(56,105,116,58)], [(1,117,25,70),(2,71,26,118),(3,119,27,72),(4,65,28,120),(5,113,29,66),(6,67,30,114),(7,115,31,68),(8,69,32,116),(9,18,64,38),(10,39,57,19),(11,20,58,40),(12,33,59,21),(13,22,60,34),(14,35,61,23),(15,24,62,36),(16,37,63,17),(41,74,81,110),(42,111,82,75),(43,76,83,112),(44,105,84,77),(45,78,85,106),(46,107,86,79),(47,80,87,108),(48,109,88,73),(49,102,125,95),(50,96,126,103),(51,104,127,89),(52,90,128,97),(53,98,121,91),(54,92,122,99),(55,100,123,93),(56,94,124,101)], [(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···4X4Y···4AN8A···8AF
order12···24···44···48···8
size11···11···12···22···2

80 irreducible representations

dim111111112222
type++++++-
imageC1C2C2C2C2C4C4C8D4Q8M4(2)C4○D4
kernelC4×C4⋊C8C22.7C42C43C2×C4×C8C2×C4⋊C8C4⋊C8C2×C42C42C42C42C2×C4C2×C4
# reps12122168322284

Matrix representation of C4×C4⋊C8 in GL4(𝔽17) generated by

4000
01300
00130
00013
,
16000
0100
0004
0040
,
2000
0400
0090
0008
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0],[2,0,0,0,0,4,0,0,0,0,9,0,0,0,0,8] >;

C4×C4⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes C_8
% in TeX

G:=Group("C4xC4:C8");
// GroupNames label

G:=SmallGroup(128,498);
// by ID

G=gap.SmallGroup(128,498);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,124]);
// Polycyclic

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

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