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G = C42.201D4order 128 = 27

183rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.201D4, C23.722C24, C22.4952+ 1+4, C22.3782- 1+4, C428C4.50C2, (C2×C42).734C22, (C22×C4).233C23, C22.454(C22×D4), C2.C42.425C22, C23.83C23.47C2, C23.81C23.50C2, C2.6(C22.58C24), C2.71(C23.38C23), C2.61(C22.57C24), C2.60(C22.31C24), (C2×C4).439(C2×D4), (C2×C4⋊C4).531C22, (C2×C42.C2).29C2, SmallGroup(128,1554)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.201D4
C1C2C22C23C22×C4C2×C4⋊C4C23.83C23 — C42.201D4
C1C23 — C42.201D4
C1C23 — C42.201D4
C1C23 — C42.201D4

Generators and relations for C42.201D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 324 in 190 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×6], C2×C4 [×42], C23, C42 [×4], C4⋊C4 [×30], C22×C4, C22×C4 [×14], C2.C42 [×16], C2×C42, C2×C4⋊C4 [×18], C42.C2 [×8], C428C4, C23.81C23 [×8], C23.83C23 [×4], C2×C42.C2 [×2], C42.201D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4 [×5], C23.38C23 [×2], C22.31C24, C22.57C24 [×2], C22.58C24 [×2], C42.201D4

Character table of C42.201D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-111-1-11111-1-1-1-111-1-1    linear of order 2
ρ31111111111-1-1-1-1-111-11-11-1-11-11    linear of order 2
ρ411111111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ511111111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ611111111-1-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ71111111111-1-1-1-11-11-1-11-111-1-11    linear of order 2
ρ811111111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111111-1-111-1-111-1-1-1-111-1-111    linear of order 2
ρ111111111111-1-1-1-1-11-111-1-111-11-1    linear of order 2
ρ1211111111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ1311111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1411111111-1-111-1-1-1-1-1-111-1-11111    linear of order 2
ρ151111111111-1-1-1-11-1-11-111-1-111-1    linear of order 2
ρ1611111111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ172-22-22-22-2-222-2-22000000000000    orthogonal lifted from D4
ρ182-22-22-22-22-22-22-2000000000000    orthogonal lifted from D4
ρ192-22-22-22-2-22-222-2000000000000    orthogonal lifted from D4
ρ202-22-22-22-22-2-22-22000000000000    orthogonal lifted from D4
ρ2144-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ234-444-4-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ254-4-4-444-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-444-4-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.201D4
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 76 103 13)(2 73 104 14)(3 74 101 15)(4 75 102 16)(5 36 70 95)(6 33 71 96)(7 34 72 93)(8 35 69 94)(9 107 40 48)(10 108 37 45)(11 105 38 46)(12 106 39 47)(17 56 80 115)(18 53 77 116)(19 54 78 113)(20 55 79 114)(21 111 84 52)(22 112 81 49)(23 109 82 50)(24 110 83 51)(25 121 88 62)(26 122 85 63)(27 123 86 64)(28 124 87 61)(29 58 92 117)(30 59 89 118)(31 60 90 119)(32 57 91 120)(41 126 97 68)(42 127 98 65)(43 128 99 66)(44 125 100 67)
(1 33 9 65)(2 36 10 68)(3 35 11 67)(4 34 12 66)(5 106 97 75)(6 105 98 74)(7 108 99 73)(8 107 100 76)(13 69 48 44)(14 72 45 43)(15 71 46 42)(16 70 47 41)(17 117 52 86)(18 120 49 85)(19 119 50 88)(20 118 51 87)(21 121 56 90)(22 124 53 89)(23 123 54 92)(24 122 55 91)(25 78 60 109)(26 77 57 112)(27 80 58 111)(28 79 59 110)(29 82 64 113)(30 81 61 116)(31 84 62 115)(32 83 63 114)(37 126 104 95)(38 125 101 94)(39 128 102 93)(40 127 103 96)
(1 119 103 60)(2 57 104 120)(3 117 101 58)(4 59 102 118)(5 53 70 116)(6 113 71 54)(7 55 72 114)(8 115 69 56)(9 88 40 25)(10 26 37 85)(11 86 38 27)(12 28 39 87)(13 92 76 29)(14 30 73 89)(15 90 74 31)(16 32 75 91)(17 96 80 33)(18 34 77 93)(19 94 78 35)(20 36 79 95)(21 100 84 44)(22 41 81 97)(23 98 82 42)(24 43 83 99)(45 61 108 124)(46 121 105 62)(47 63 106 122)(48 123 107 64)(49 66 112 128)(50 125 109 67)(51 68 110 126)(52 127 111 65)

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

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

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

Matrix representation of C42.201D4 in GL12(𝔽5)

001000000000
000100000000
400000000000
040000000000
000021000000
000003000000
000014030000
000014300000
000000000034
000000002413
000000000423
000000004214
,
100000000000
040000000000
001000000000
000400000000
000042000000
000001000000
000003040000
000003400000
000000001100
000000003400
000000004204
000000000210
,
030000000000
300000000000
000200000000
002000000000
000034000000
000032000000
000041020000
000001300000
000000003404
000000002411
000000000422
000000001341
,
010000000000
100000000000
000100000000
001000000000
000040040000
000040220000
000023010000
000000010000
000000001431
000000004131
000000000211
000000002122

G:=sub<GL(12,GF(5))| [0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,1,0,0,0,0,0,0,0,0,1,3,4,4,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,0,3,1,2,1,0,0,0,0,0,0,0,0,4,3,3,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,1,3,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,4,0,0,0,0,0,0,0,0,0,1,4,2,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0],[0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,4,0,0,0,0,0,0,0,0,0,4,2,1,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,0,1,0,0,0,0,0,0,0,0,4,4,4,3,0,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,4,1,2,1],[0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,0,2,0,0,0,0,0,0,0,0,4,1,2,1,0,0,0,0,0,0,0,0,3,3,1,2,0,0,0,0,0,0,0,0,1,1,1,2] >;

C42.201D4 in GAP, Magma, Sage, TeX

C_4^2._{201}D_4
% in TeX

G:=Group("C4^2.201D4");
// GroupNames label

G:=SmallGroup(128,1554);
// by ID

G=gap.SmallGroup(128,1554);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,184,794,185,80]);
// Polycyclic

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

Export

Character table of C42.201D4 in TeX

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