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G = C43.7C2order 128 = 27

7th non-split extension by C43 of C2 acting faithfully

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

Aliases: C43.7C2, C42.44Q8, C42.307D4, C4⋊C811C4, C41(C8⋊C4), C4.42(C4×Q8), C4.166(C4×D4), (C2×C4).47C42, (C2×C42).35C4, C42.264(C2×C4), (C2×C4).76M4(2), C2.12(C4×M4(2)), C22.53(C2×C42), C2.2(C4⋊M4(2)), C2.3(C42.6C4), (C22×C8).383C22, (C2×C42).993C22, C23.257(C22×C4), C22.41(C2×M4(2)), (C22×C4).1610C23, C22.52(C42⋊C2), C22.7C42.40C2, C2.7(C4×C4⋊C4), (C2×C4⋊C8).51C2, C2.9(C2×C8⋊C4), (C2×C4).78(C4⋊C4), (C2×C8).135(C2×C4), C22.56(C2×C4⋊C4), (C2×C4).330(C2×Q8), (C2×C8⋊C4).24C2, (C2×C4).1502(C2×D4), (C2×C4).920(C4○D4), (C2×C4).600(C22×C4), (C22×C4).439(C2×C4), SmallGroup(128,499)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C43.7C2
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — C43.7C2
C1C22 — C43.7C2
C1C22×C4 — C43.7C2
C1C2C2C22×C4 — C43.7C2

Generators and relations for C43.7C2
 G = < a,b,c,d | a4=b4=c4=1, d2=c, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=bc2, cd=dc >

Subgroups: 196 in 146 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×10], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×20], C2×C4 [×10], C23, C42 [×8], C42 [×10], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×4], C8⋊C4 [×4], C4⋊C8 [×8], C2×C42 [×3], C2×C42 [×4], C22×C8 [×4], C22.7C42 [×2], C43, C2×C8⋊C4 [×2], C2×C4⋊C8 [×2], C43.7C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], M4(2) [×8], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C8⋊C4 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×4], C4×C4⋊C4, C2×C8⋊C4, C4×M4(2), C4⋊M4(2) [×2], C42.6C4 [×2], C43.7C2

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

dim11111112222
type++++++-
imageC1C2C2C2C2C4C4D4Q8M4(2)C4○D4
kernelC43.7C2C22.7C42C43C2×C8⋊C4C2×C4⋊C8C4⋊C8C2×C42C42C42C2×C4C2×C4
# reps1212216822164

Matrix representation of C43.7C2 in GL5(𝔽17)

130000
04000
00400
000013
00040
,
40000
011500
011600
00001
000160
,
160000
04000
00400
000130
000013
,
40000
00500
011000
000315
0001514

G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,13,0],[4,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,13],[4,0,0,0,0,0,0,11,0,0,0,5,0,0,0,0,0,0,3,15,0,0,0,15,14] >;

C43.7C2 in GAP, Magma, Sage, TeX

C_4^3._7C_2
% in TeX

G:=Group("C4^3.7C2");
// GroupNames label

G:=SmallGroup(128,499);
// by ID

G=gap.SmallGroup(128,499);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,142,124]);
// 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^-1*b^2,b*c=c*b,d*b*d^-1=b*c^2,c*d=d*c>;
// generators/relations

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