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G = C4⋊C43C8order 128 = 27

2nd semidirect product of C4⋊C4 and C8 acting via C8/C4=C2

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

Aliases: C4⋊C43C8, C2.3(C8×Q8), C2.12(C8×D4), (C2×C8).38Q8, (C2×C8).325D4, C2.6(C89D4), C2.3(C84Q8), C22.25(C4×Q8), C22.100(C4×D4), (C2×C4).36M4(2), C22.31(C8○D4), C4.114(C22⋊Q8), (C22×C8).40C22, C22.41(C22×C8), C4.36(C42.C2), C4.46(C422C2), C2.C42.21C4, (C2×C42).290C22, C23.270(C22×C4), C22.52(C2×M4(2)), C2.9(C42.12C4), (C22×C4).1631C23, C22.7C42.7C2, C22.58(C42⋊C2), C4.137(C22.D4), C2.4(C42.7C22), C2.3(C23.63C23), (C2×C4×C8).20C2, (C2×C4⋊C8).28C2, (C4×C4⋊C4).12C2, (C2×C4⋊C4).56C4, (C2×C4).20(C2×C8), (C2×C4).343(C2×Q8), (C2×C4).1532(C2×D4), (C2×C4).937(C4○D4), (C22×C4).121(C2×C4), SmallGroup(128,648)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊C43C8
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C4⋊C43C8
C1C22 — C4⋊C43C8
C1C22×C4 — C4⋊C43C8
C1C2C2C22×C4 — C4⋊C43C8

Generators and relations for C4⋊C43C8
 G = < a,b,c | a4=b4=c8=1, bab-1=a-1, cac-1=ab2, bc=cb >

Subgroups: 180 in 120 conjugacy classes, 68 normal (52 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2.C42, C4×C8, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C4⋊C43C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C22×C8, C2×M4(2), C8○D4, C23.63C23, C42.12C4, C42.7C22, C8×D4, C89D4, C8×Q8, C84Q8, C4⋊C43C8

Smallest permutation representation of C4⋊C43C8
Regular action on 128 points
Generators in S128
(1 40 55 97)(2 125 56 106)(3 34 49 99)(4 127 50 108)(5 36 51 101)(6 121 52 110)(7 38 53 103)(8 123 54 112)(9 37 28 102)(10 122 29 111)(11 39 30 104)(12 124 31 105)(13 33 32 98)(14 126 25 107)(15 35 26 100)(16 128 27 109)(17 113 76 68)(18 82 77 92)(19 115 78 70)(20 84 79 94)(21 117 80 72)(22 86 73 96)(23 119 74 66)(24 88 75 90)(41 116 63 71)(42 85 64 95)(43 118 57 65)(44 87 58 89)(45 120 59 67)(46 81 60 91)(47 114 61 69)(48 83 62 93)
(1 24 12 45)(2 17 13 46)(3 18 14 47)(4 19 15 48)(5 20 16 41)(6 21 9 42)(7 22 10 43)(8 23 11 44)(25 61 49 77)(26 62 50 78)(27 63 51 79)(28 64 52 80)(29 57 53 73)(30 58 54 74)(31 59 55 75)(32 60 56 76)(33 91 125 68)(34 92 126 69)(35 93 127 70)(36 94 128 71)(37 95 121 72)(38 96 122 65)(39 89 123 66)(40 90 124 67)(81 106 113 98)(82 107 114 99)(83 108 115 100)(84 109 116 101)(85 110 117 102)(86 111 118 103)(87 112 119 104)(88 105 120 97)
(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)| (1,40,55,97)(2,125,56,106)(3,34,49,99)(4,127,50,108)(5,36,51,101)(6,121,52,110)(7,38,53,103)(8,123,54,112)(9,37,28,102)(10,122,29,111)(11,39,30,104)(12,124,31,105)(13,33,32,98)(14,126,25,107)(15,35,26,100)(16,128,27,109)(17,113,76,68)(18,82,77,92)(19,115,78,70)(20,84,79,94)(21,117,80,72)(22,86,73,96)(23,119,74,66)(24,88,75,90)(41,116,63,71)(42,85,64,95)(43,118,57,65)(44,87,58,89)(45,120,59,67)(46,81,60,91)(47,114,61,69)(48,83,62,93), (1,24,12,45)(2,17,13,46)(3,18,14,47)(4,19,15,48)(5,20,16,41)(6,21,9,42)(7,22,10,43)(8,23,11,44)(25,61,49,77)(26,62,50,78)(27,63,51,79)(28,64,52,80)(29,57,53,73)(30,58,54,74)(31,59,55,75)(32,60,56,76)(33,91,125,68)(34,92,126,69)(35,93,127,70)(36,94,128,71)(37,95,121,72)(38,96,122,65)(39,89,123,66)(40,90,124,67)(81,106,113,98)(82,107,114,99)(83,108,115,100)(84,109,116,101)(85,110,117,102)(86,111,118,103)(87,112,119,104)(88,105,120,97), (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( (1,40,55,97)(2,125,56,106)(3,34,49,99)(4,127,50,108)(5,36,51,101)(6,121,52,110)(7,38,53,103)(8,123,54,112)(9,37,28,102)(10,122,29,111)(11,39,30,104)(12,124,31,105)(13,33,32,98)(14,126,25,107)(15,35,26,100)(16,128,27,109)(17,113,76,68)(18,82,77,92)(19,115,78,70)(20,84,79,94)(21,117,80,72)(22,86,73,96)(23,119,74,66)(24,88,75,90)(41,116,63,71)(42,85,64,95)(43,118,57,65)(44,87,58,89)(45,120,59,67)(46,81,60,91)(47,114,61,69)(48,83,62,93), (1,24,12,45)(2,17,13,46)(3,18,14,47)(4,19,15,48)(5,20,16,41)(6,21,9,42)(7,22,10,43)(8,23,11,44)(25,61,49,77)(26,62,50,78)(27,63,51,79)(28,64,52,80)(29,57,53,73)(30,58,54,74)(31,59,55,75)(32,60,56,76)(33,91,125,68)(34,92,126,69)(35,93,127,70)(36,94,128,71)(37,95,121,72)(38,96,122,65)(39,89,123,66)(40,90,124,67)(81,106,113,98)(82,107,114,99)(83,108,115,100)(84,109,116,101)(85,110,117,102)(86,111,118,103)(87,112,119,104)(88,105,120,97), (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([[(1,40,55,97),(2,125,56,106),(3,34,49,99),(4,127,50,108),(5,36,51,101),(6,121,52,110),(7,38,53,103),(8,123,54,112),(9,37,28,102),(10,122,29,111),(11,39,30,104),(12,124,31,105),(13,33,32,98),(14,126,25,107),(15,35,26,100),(16,128,27,109),(17,113,76,68),(18,82,77,92),(19,115,78,70),(20,84,79,94),(21,117,80,72),(22,86,73,96),(23,119,74,66),(24,88,75,90),(41,116,63,71),(42,85,64,95),(43,118,57,65),(44,87,58,89),(45,120,59,67),(46,81,60,91),(47,114,61,69),(48,83,62,93)], [(1,24,12,45),(2,17,13,46),(3,18,14,47),(4,19,15,48),(5,20,16,41),(6,21,9,42),(7,22,10,43),(8,23,11,44),(25,61,49,77),(26,62,50,78),(27,63,51,79),(28,64,52,80),(29,57,53,73),(30,58,54,74),(31,59,55,75),(32,60,56,76),(33,91,125,68),(34,92,126,69),(35,93,127,70),(36,94,128,71),(37,95,121,72),(38,96,122,65),(39,89,123,66),(40,90,124,67),(81,106,113,98),(82,107,114,99),(83,108,115,100),(84,109,116,101),(85,110,117,102),(86,111,118,103),(87,112,119,104),(88,105,120,97)], [(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)]])

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim1111111122222
type++++++-
imageC1C2C2C2C2C4C4C8D4Q8M4(2)C4○D4C8○D4
kernelC4⋊C43C8C22.7C42C4×C4⋊C4C2×C4×C8C2×C4⋊C8C2.C42C2×C4⋊C4C4⋊C4C2×C8C2×C8C2×C4C2×C4C22
# reps14111441622488

Matrix representation of C4⋊C43C8 in GL5(𝔽17)

160000
011900
015600
000112
00076
,
160000
013000
001300
00042
000013
,
20000
00100
04000
000168
00001

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,11,15,0,0,0,9,6,0,0,0,0,0,11,7,0,0,0,2,6],[16,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,2,13],[2,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,8,1] >;

C4⋊C43C8 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_3C_8
% in TeX

G:=Group("C4:C4:3C8");
// GroupNames label

G:=SmallGroup(128,648);
// by ID

G=gap.SmallGroup(128,648);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,124]);
// Polycyclic

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

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