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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 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], C23, C42 [×6], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], C22×C4 [×7], C2.C42 [×2], C4×C8 [×2], C4⋊C8 [×2], C2×C42 [×3], C2×C4⋊C4 [×2], C22×C8 [×4], C22.7C42 [×4], C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C4⋊C43C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C22×C8, C2×M4(2), C8○D4 [×2], 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 68)(3 34 49 99)(4 127 50 70)(5 36 51 101)(6 121 52 72)(7 38 53 103)(8 123 54 66)(9 87 58 107)(10 120 59 92)(11 81 60 109)(12 114 61 94)(13 83 62 111)(14 116 63 96)(15 85 64 105)(16 118 57 90)(17 122 29 65)(18 39 30 104)(19 124 31 67)(20 33 32 98)(21 126 25 69)(22 35 26 100)(23 128 27 71)(24 37 28 102)(41 88 75 108)(42 113 76 93)(43 82 77 110)(44 115 78 95)(45 84 79 112)(46 117 80 89)(47 86 73 106)(48 119 74 91)
(1 41 19 10)(2 42 20 11)(3 43 21 12)(4 44 22 13)(5 45 23 14)(6 46 24 15)(7 47 17 16)(8 48 18 9)(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)(33 109 125 93)(34 110 126 94)(35 111 127 95)(36 112 128 96)(37 105 121 89)(38 106 122 90)(39 107 123 91)(40 108 124 92)(65 118 103 86)(66 119 104 87)(67 120 97 88)(68 113 98 81)(69 114 99 82)(70 115 100 83)(71 116 101 84)(72 117 102 85)
(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)(2,125,56,68)(3,34,49,99)(4,127,50,70)(5,36,51,101)(6,121,52,72)(7,38,53,103)(8,123,54,66)(9,87,58,107)(10,120,59,92)(11,81,60,109)(12,114,61,94)(13,83,62,111)(14,116,63,96)(15,85,64,105)(16,118,57,90)(17,122,29,65)(18,39,30,104)(19,124,31,67)(20,33,32,98)(21,126,25,69)(22,35,26,100)(23,128,27,71)(24,37,28,102)(41,88,75,108)(42,113,76,93)(43,82,77,110)(44,115,78,95)(45,84,79,112)(46,117,80,89)(47,86,73,106)(48,119,74,91), (1,41,19,10)(2,42,20,11)(3,43,21,12)(4,44,22,13)(5,45,23,14)(6,46,24,15)(7,47,17,16)(8,48,18,9)(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)(33,109,125,93)(34,110,126,94)(35,111,127,95)(36,112,128,96)(37,105,121,89)(38,106,122,90)(39,107,123,91)(40,108,124,92)(65,118,103,86)(66,119,104,87)(67,120,97,88)(68,113,98,81)(69,114,99,82)(70,115,100,83)(71,116,101,84)(72,117,102,85), (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)(2,125,56,68)(3,34,49,99)(4,127,50,70)(5,36,51,101)(6,121,52,72)(7,38,53,103)(8,123,54,66)(9,87,58,107)(10,120,59,92)(11,81,60,109)(12,114,61,94)(13,83,62,111)(14,116,63,96)(15,85,64,105)(16,118,57,90)(17,122,29,65)(18,39,30,104)(19,124,31,67)(20,33,32,98)(21,126,25,69)(22,35,26,100)(23,128,27,71)(24,37,28,102)(41,88,75,108)(42,113,76,93)(43,82,77,110)(44,115,78,95)(45,84,79,112)(46,117,80,89)(47,86,73,106)(48,119,74,91), (1,41,19,10)(2,42,20,11)(3,43,21,12)(4,44,22,13)(5,45,23,14)(6,46,24,15)(7,47,17,16)(8,48,18,9)(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)(33,109,125,93)(34,110,126,94)(35,111,127,95)(36,112,128,96)(37,105,121,89)(38,106,122,90)(39,107,123,91)(40,108,124,92)(65,118,103,86)(66,119,104,87)(67,120,97,88)(68,113,98,81)(69,114,99,82)(70,115,100,83)(71,116,101,84)(72,117,102,85), (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),(2,125,56,68),(3,34,49,99),(4,127,50,70),(5,36,51,101),(6,121,52,72),(7,38,53,103),(8,123,54,66),(9,87,58,107),(10,120,59,92),(11,81,60,109),(12,114,61,94),(13,83,62,111),(14,116,63,96),(15,85,64,105),(16,118,57,90),(17,122,29,65),(18,39,30,104),(19,124,31,67),(20,33,32,98),(21,126,25,69),(22,35,26,100),(23,128,27,71),(24,37,28,102),(41,88,75,108),(42,113,76,93),(43,82,77,110),(44,115,78,95),(45,84,79,112),(46,117,80,89),(47,86,73,106),(48,119,74,91)], [(1,41,19,10),(2,42,20,11),(3,43,21,12),(4,44,22,13),(5,45,23,14),(6,46,24,15),(7,47,17,16),(8,48,18,9),(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),(33,109,125,93),(34,110,126,94),(35,111,127,95),(36,112,128,96),(37,105,121,89),(38,106,122,90),(39,107,123,91),(40,108,124,92),(65,118,103,86),(66,119,104,87),(67,120,97,88),(68,113,98,81),(69,114,99,82),(70,115,100,83),(71,116,101,84),(72,117,102,85)], [(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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