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G = C42.161D4order 128 = 27

143rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.161D4, C23.209C24, C22.322- 1+4, C4⋊Q830C4, C4.38(C4×D4), C42.180(C2×C4), C428C4.20C2, C424C4.13C2, C22.97(C22×D4), (C22×C4).474C23, (C2×C42).416C22, C22.100(C23×C4), (C22×Q8).400C22, C2.C42.45C22, C23.63C23.2C2, C23.67C23.25C2, C2.4(C22.35C24), C2.9(C23.32C23), C2.7(C23.38C23), C2.26(C2×C4×D4), (C2×C4×Q8).22C2, (C2×C4⋊Q8).24C2, C4⋊C4.104(C2×C4), (C2×C4).1189(C2×D4), (C2×C4).30(C22×C4), (C2×Q8).149(C2×C4), C22.94(C2×C4○D4), (C2×C4).649(C4○D4), (C2×C4⋊C4).808C22, SmallGroup(128,1059)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.161D4
C1C2C22C23C22×C4C2×C42C424C4 — C42.161D4
C1C22 — C42.161D4
C1C23 — C42.161D4
C1C23 — C42.161D4

Generators and relations for C42.161D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 380 in 258 conjugacy classes, 148 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×16], C23, C42 [×8], C42 [×10], C4⋊C4 [×16], C4⋊C4 [×12], C22×C4, C22×C4 [×14], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×16], C2×C42 [×3], C2×C42 [×4], C2×C4⋊C4 [×10], C4×Q8 [×8], C4⋊Q8 [×8], C22×Q8 [×2], C424C4, C428C4, C23.63C23 [×8], C23.67C23 [×2], C2×C4×Q8 [×2], C2×C4⋊Q8, C42.161D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2- 1+4 [×4], C2×C4×D4, C23.32C23 [×2], C23.38C23 [×2], C22.35C24 [×2], C42.161D4

Smallest permutation representation of C42.161D4
Regular action on 128 points
Generators in S128
(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 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 46 38 18)(6 47 39 19)(7 48 40 20)(8 45 37 17)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 31 23 47)(2 60 24 20)(3 29 21 45)(4 58 22 18)(5 14 70 54)(6 43 71 27)(7 16 72 56)(8 41 69 25)(9 57 49 17)(10 30 50 46)(11 59 51 19)(12 32 52 48)(13 61 53 37)(15 63 55 39)(26 38 42 62)(28 40 44 64)(33 107 65 119)(34 76 66 96)(35 105 67 117)(36 74 68 94)(73 98 93 125)(75 100 95 127)(77 109 81 121)(78 86 82 90)(79 111 83 123)(80 88 84 92)(85 113 89 101)(87 115 91 103)(97 120 128 108)(99 118 126 106)(102 110 114 122)(104 112 116 124)
(1 121 9 91)(2 122 10 92)(3 123 11 89)(4 124 12 90)(5 108 40 74)(6 105 37 75)(7 106 38 76)(8 107 39 73)(13 33 43 125)(14 34 44 126)(15 35 41 127)(16 36 42 128)(17 79 47 101)(18 80 48 102)(19 77 45 103)(20 78 46 104)(21 111 51 85)(22 112 52 86)(23 109 49 87)(24 110 50 88)(25 100 55 67)(26 97 56 68)(27 98 53 65)(28 99 54 66)(29 115 59 81)(30 116 60 82)(31 113 57 83)(32 114 58 84)(61 95 71 117)(62 96 72 118)(63 93 69 119)(64 94 70 120)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4AJ
order12···24···44···4
size11···12···24···4

44 irreducible representations

dim11111111224
type++++++++-
imageC1C2C2C2C2C2C2C4D4C4○D42- 1+4
kernelC42.161D4C424C4C428C4C23.63C23C23.67C23C2×C4×Q8C2×C4⋊Q8C4⋊Q8C42C2×C4C22
# reps111822116444

Matrix representation of C42.161D4 in GL8(𝔽5)

20000000
02000000
00100000
00010000
00001000
00000100
00000040
00003204
,
10000000
01000000
00400000
00040000
00000100
00004000
00003213
00003014
,
32000000
02000000
00330000
00020000
00000010
00003213
00001000
00001403
,
14000000
24000000
00400000
00210000
00003000
00000200
00000020
00000423

G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,3,3,0,0,0,0,1,0,2,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4],[3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,3,1,1,0,0,0,0,0,2,0,4,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,3],[1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3] >;

C42.161D4 in GAP, Magma, Sage, TeX

C_4^2._{161}D_4
% in TeX

G:=Group("C4^2.161D4");
// GroupNames label

G:=SmallGroup(128,1059);
// by ID

G=gap.SmallGroup(128,1059);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,268,675,80]);
// Polycyclic

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

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