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G = C42.385D4order 128 = 27

18th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.385D4, (C2×C8)⋊4C8, C4⋊C8.6C4, C4.2(C4×C8), (C2×C4).156D8, (C2×C4).65Q16, C2.1(C82C8), C2.1(C81C8), C4.2(C8⋊C4), (C22×C8).22C4, (C2×C4).50C42, (C22×C4).67Q8, C4.17(C22⋊C8), C22.15(C4⋊C8), C23.78(C4⋊C4), C42.251(C2×C4), (C2×C4).123SD16, (C22×C4).806D4, (C2×C4).66M4(2), C4.41(D4⋊C4), C4.27(Q8⋊C4), C22.15(C2.D8), C22.11(C4.Q8), C2.1(C22.4Q16), C22.9(C8.C4), C2.1(C4.C42), (C2×C42).1025C22, C2.6(C22.7C42), C22.18(C2.C42), (C2×C4×C8).2C2, (C2×C4⋊C8).1C2, (C2×C4).69(C2×C8), (C2×C4).64(C4⋊C4), (C22×C4).463(C2×C4), (C2×C4).369(C22⋊C4), SmallGroup(128,9)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.385D4
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C42.385D4
C1C2C4 — C42.385D4
C1C22×C4C2×C42 — C42.385D4
C1C22C22C2×C42 — C42.385D4

Generators and relations for C42.385D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 136 in 90 conjugacy classes, 60 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×4], C2×C4 [×10], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C22×C8 [×2], C2×C4×C8, C2×C4⋊C8 [×2], C42.385D4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], D8, SD16 [×2], Q16, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8, C2.D8, C8.C4 [×2], C82C8 [×2], C81C8 [×2], C22.7C42, C22.4Q16, C4.C42, C42.385D4

Smallest permutation representation of C42.385D4
Regular action on 128 points
Generators in S128
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(25 115 29 119)(26 116 30 120)(27 117 31 113)(28 118 32 114)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(41 74 45 78)(42 75 46 79)(43 76 47 80)(44 77 48 73)(65 85 69 81)(66 86 70 82)(67 87 71 83)(68 88 72 84)(89 112 93 108)(90 105 94 109)(91 106 95 110)(92 107 96 111)(97 122 101 126)(98 123 102 127)(99 124 103 128)(100 125 104 121)
(1 117 125 77)(2 78 126 118)(3 119 127 79)(4 80 128 120)(5 113 121 73)(6 74 122 114)(7 115 123 75)(8 76 124 116)(9 34 66 106)(10 107 67 35)(11 36 68 108)(12 109 69 37)(13 38 70 110)(14 111 71 39)(15 40 72 112)(16 105 65 33)(17 60 86 95)(18 96 87 61)(19 62 88 89)(20 90 81 63)(21 64 82 91)(22 92 83 57)(23 58 84 93)(24 94 85 59)(25 98 42 51)(26 52 43 99)(27 100 44 53)(28 54 45 101)(29 102 46 55)(30 56 47 103)(31 104 48 49)(32 50 41 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)
(1 67 49 87 5 71 53 83)(2 60 50 38 6 64 54 34)(3 69 51 81 7 65 55 85)(4 62 52 40 8 58 56 36)(9 78 17 41 13 74 21 45)(10 104 18 121 14 100 22 125)(11 80 19 43 15 76 23 47)(12 98 20 123 16 102 24 127)(25 90 115 105 29 94 119 109)(26 72 116 84 30 68 120 88)(27 92 117 107 31 96 113 111)(28 66 118 86 32 70 114 82)(33 46 59 79 37 42 63 75)(35 48 61 73 39 44 57 77)(89 99 112 124 93 103 108 128)(91 101 106 126 95 97 110 122)

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim11111122222222
type+++++-+-
imageC1C2C2C4C4C8D4D4Q8M4(2)D8SD16Q16C8.C4
kernelC42.385D4C2×C4×C8C2×C4⋊C8C4⋊C8C22×C8C2×C8C42C22×C4C22×C4C2×C4C2×C4C2×C4C2×C4C22
# reps112841621142428

Matrix representation of C42.385D4 in GL4(𝔽17) generated by

13000
0100
00130
00013
,
1000
01600
0001
00160
,
9000
0400
00315
001514
,
9000
01600
00214
001415
G:=sub<GL(4,GF(17))| [13,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[9,0,0,0,0,4,0,0,0,0,3,15,0,0,15,14],[9,0,0,0,0,16,0,0,0,0,2,14,0,0,14,15] >;

C42.385D4 in GAP, Magma, Sage, TeX

C_4^2._{385}D_4
% in TeX

G:=Group("C4^2.385D4");
// GroupNames label

G:=SmallGroup(128,9);
// by ID

G=gap.SmallGroup(128,9);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684,102]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;
// generators/relations

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