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G = C42.255C23order 128 = 27

116th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.255C23, C4⋊C4.77D4, (C2×C8).193D4, (C2×Q8).64D4, C84Q8.9C2, C82C8.11C2, C4⋊C8.37C22, C4.Q16.8C2, C4⋊Q8.76C22, C4.110(C4○D8), (C4×C8).288C22, Q8⋊Q8.11C2, C4.10D8.9C2, C4.6Q16.8C2, (C4×Q8).52C22, C2.11(C8.D4), C4.47(C8.C22), C4.SD16.13C2, C2.23(D4.3D4), C2.15(Q8.D4), C22.216(C4⋊D4), (C2×C4).40(C4○D4), (C2×C4).1290(C2×D4), SmallGroup(128,436)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.255C23
C1C2C22C2×C4C42C4×Q8C84Q8 — C42.255C23
C1C22C42 — C42.255C23
C1C22C42 — C42.255C23
C1C22C22C42 — C42.255C23

Generators and relations for C42.255C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a-1b2, e2=b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a-1c, ece-1=bc, ede-1=a2d >

Subgroups: 144 in 70 conjugacy classes, 32 normal (all characteristic)
C1, C2 [×3], C4 [×4], C4 [×5], C22, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×5], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4, Q8⋊C4 [×4], C4⋊C8 [×3], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C4⋊Q8 [×2], C4.10D8, C4.6Q16, C82C8, C84Q8, Q8⋊Q8, C4.Q16, C4.SD16, C42.255C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C4○D8, C8.C22 [×3], Q8.D4, C8.D4, D4.3D4, C42.255C23

Character table of C42.255C23

 class 12A2B2C4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J
 size 1111222248816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ3111111111-1-1-11-1-1-1-1-11-1111    linear of order 2
ρ4111111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ511111111111-1-11111-1-1-111-1    linear of order 2
ρ6111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ7111111111-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ8111111111-1-1-1-11111111-1-11    linear of order 2
ρ92222-22-22-22-2000000000000    orthogonal lifted from D4
ρ102222-22-22-2-22000000000000    orthogonal lifted from D4
ρ1122222-22-2-200002-2-22000000    orthogonal lifted from D4
ρ1222222-22-2-20000-222-2000000    orthogonal lifted from D4
ρ132222-2-2-2-22000000000002i-2i0    complex lifted from C4○D4
ρ142222-2-2-2-2200000000000-2i2i0    complex lifted from C4○D4
ρ152-2-22-2020000000-2i2i02--2-200-2    complex lifted from C4○D8
ρ162-2-22-20200000002i-2i0-2--2200-2    complex lifted from C4○D8
ρ172-2-22-2020000000-2i2i0-2-2200--2    complex lifted from C4○D8
ρ182-2-22-20200000002i-2i02-2-200--2    complex lifted from C4○D8
ρ194-4-4440-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-44-40-404000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-44-4040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002-200-2-2000000    complex lifted from D4.3D4
ρ2344-4-4000000000-2-2002-2000000    complex lifted from D4.3D4

Smallest permutation representation of C42.255C23
Regular action on 128 points
Generators in S128
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 41 13 45)(10 42 14 46)(11 43 15 47)(12 44 16 48)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 122 37 126)(34 123 38 127)(35 124 39 128)(36 125 40 121)(57 74 61 78)(58 75 62 79)(59 76 63 80)(60 77 64 73)(65 104 69 100)(66 97 70 101)(67 98 71 102)(68 99 72 103)(81 94 85 90)(82 95 86 91)(83 96 87 92)(84 89 88 93)(105 113 109 117)(106 114 110 118)(107 115 111 119)(108 116 112 120)
(1 61 18 80)(2 62 19 73)(3 63 20 74)(4 64 21 75)(5 57 22 76)(6 58 23 77)(7 59 24 78)(8 60 17 79)(9 118 43 108)(10 119 44 109)(11 120 45 110)(12 113 46 111)(13 114 47 112)(14 115 48 105)(15 116 41 106)(16 117 42 107)(25 104 51 71)(26 97 52 72)(27 98 53 65)(28 99 54 66)(29 100 55 67)(30 101 56 68)(31 102 49 69)(32 103 50 70)(33 95 124 88)(34 96 125 81)(35 89 126 82)(36 90 127 83)(37 91 128 84)(38 92 121 85)(39 93 122 86)(40 94 123 87)
(1 90 5 94)(2 82 6 86)(3 96 7 92)(4 88 8 84)(9 71 13 67)(10 103 14 99)(11 69 15 65)(12 101 16 97)(17 91 21 95)(18 83 22 87)(19 89 23 93)(20 81 24 85)(25 112 29 108)(26 113 30 117)(27 110 31 106)(28 119 32 115)(33 79 37 75)(34 59 38 63)(35 77 39 73)(36 57 40 61)(41 98 45 102)(42 72 46 68)(43 104 47 100)(44 70 48 66)(49 116 53 120)(50 105 54 109)(51 114 55 118)(52 111 56 107)(58 122 62 126)(60 128 64 124)(74 125 78 121)(76 123 80 127)
(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 29 18 55)(2 26 19 52)(3 31 20 49)(4 28 21 54)(5 25 22 51)(6 30 23 56)(7 27 24 53)(8 32 17 50)(9 90 43 83)(10 95 44 88)(11 92 45 85)(12 89 46 82)(13 94 47 87)(14 91 48 84)(15 96 41 81)(16 93 42 86)(33 109 124 119)(34 106 125 116)(35 111 126 113)(36 108 127 118)(37 105 128 115)(38 110 121 120)(39 107 122 117)(40 112 123 114)(57 71 76 104)(58 68 77 101)(59 65 78 98)(60 70 79 103)(61 67 80 100)(62 72 73 97)(63 69 74 102)(64 66 75 99)

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

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

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

Matrix representation of C42.255C23 in GL6(𝔽17)

100000
010000
0011500
0011600
0001601
00116160
,
1150000
1160000
001000
000100
000010
000001
,
1150000
0160000
004131313
0000013
0000130
0001300
,
1300000
0130000
007101210
007111116
00710010
00011116
,
070000
1200000
0010150
0000161
0000160
0001160

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,0,0,0,0,0,13,0,0,13,0,0,13,0,13,0,0,0,13,13,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,7,7,0,0,0,10,11,10,11,0,0,12,11,0,1,0,0,10,16,10,16],[0,12,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0] >;

C42.255C23 in GAP, Magma, Sage, TeX

C_4^2._{255}C_2^3
% in TeX

G:=Group("C4^2.255C2^3");
// GroupNames label

G:=SmallGroup(128,436);
// by ID

G=gap.SmallGroup(128,436);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C42.255C23 in TeX

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