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G = C43.C2order 128 = 27

3rd non-split extension by C43 of C2 acting faithfully

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

Aliases: C43.3C2, C8⋊C412C4, (C2×C4).41C42, (C2×C42).31C4, C2.9(C4×M4(2)), C42.243(C2×C4), (C2×C4).58M4(2), C2.4(C424C4), C22.47(C2×C42), C4.68(C42⋊C2), C2.1(C42.6C4), (C22×C8).374C22, C23.248(C22×C4), (C2×C42).985C22, C22.35(C2×M4(2)), (C22×C4).1601C23, C22.46(C42⋊C2), C22.7C42.38C2, (C2×C8).128(C2×C4), (C2×C8⋊C4).21C2, (C2×C4).911(C4○D4), (C22×C4).432(C2×C4), (C2×C4).591(C22×C4), SmallGroup(128,477)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C43.C2
C1C2C22C2×C4C22×C4C2×C42C43 — C43.C2
C1C22 — C43.C2
C1C22×C4 — C43.C2
C1C2C2C22×C4 — C43.C2

Generators and relations for C43.C2
 G = < a,b,c,d | a4=b4=c4=1, d2=c, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=bc2, cd=dc >

Subgroups: 188 in 136 conjugacy classes, 84 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×18], C2×C4 [×12], C23, C42 [×4], C42 [×12], C2×C8 [×8], C2×C8 [×8], C22×C4, C22×C4 [×6], C8⋊C4 [×8], C2×C42, C2×C42 [×6], C22×C8 [×4], C22.7C42 [×4], C43, C2×C8⋊C4 [×2], C43.C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×8], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C2×M4(2) [×4], C424C4, C4×M4(2) [×2], C42.6C4 [×4], C43.C2

Smallest permutation representation of C43.C2
Regular action on 128 points
Generators in S128
(1 119 103 89)(2 64 104 34)(3 113 97 91)(4 58 98 36)(5 115 99 93)(6 60 100 38)(7 117 101 95)(8 62 102 40)(9 32 126 54)(10 81 127 111)(11 26 128 56)(12 83 121 105)(13 28 122 50)(14 85 123 107)(15 30 124 52)(16 87 125 109)(17 63 47 33)(18 120 48 90)(19 57 41 35)(20 114 42 92)(21 59 43 37)(22 116 44 94)(23 61 45 39)(24 118 46 96)(25 69 55 79)(27 71 49 73)(29 65 51 75)(31 67 53 77)(66 108 76 86)(68 110 78 88)(70 112 80 82)(72 106 74 84)
(1 25 17 81)(2 30 18 86)(3 27 19 83)(4 32 20 88)(5 29 21 85)(6 26 22 82)(7 31 23 87)(8 28 24 84)(9 92 78 36)(10 89 79 33)(11 94 80 38)(12 91 73 35)(13 96 74 40)(14 93 75 37)(15 90 76 34)(16 95 77 39)(41 105 97 49)(42 110 98 54)(43 107 99 51)(44 112 100 56)(45 109 101 53)(46 106 102 50)(47 111 103 55)(48 108 104 52)(57 121 113 71)(58 126 114 68)(59 123 115 65)(60 128 116 70)(61 125 117 67)(62 122 118 72)(63 127 119 69)(64 124 120 66)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4AF8A···8P
order12···24···44···48···8
size11···11···12···24···4

56 irreducible representations

dim11111122
type++++
imageC1C2C2C2C4C4M4(2)C4○D4
kernelC43.C2C22.7C42C43C2×C8⋊C4C8⋊C4C2×C42C2×C4C2×C4
# reps1412168168

Matrix representation of C43.C2 in GL5(𝔽17)

40000
00100
016000
00010
00001
,
10000
00100
016000
00001
00010
,
10000
04000
00400
00040
00004
,
160000
061100
0111100
000814
00039

G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,6,11,0,0,0,11,11,0,0,0,0,0,8,3,0,0,0,14,9] >;

C43.C2 in GAP, Magma, Sage, TeX

C_4^3.C_2
% in TeX

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

G:=SmallGroup(128,477);
// by ID

G=gap.SmallGroup(128,477);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,58,172]);
// Polycyclic

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

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