p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43.7C2, C42.44Q8, C42.307D4, C4⋊C8⋊11C4, C4⋊1(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
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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C4⋊C8, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C8⋊C4, C2×C4⋊C8, C43.7C2
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C4×C4⋊C4, C2×C8⋊C4, C4×M4(2), C4⋊M4(2), C42.6C4, C43.7C2
►Regular action on 128 points
Generators in S128
(1 87 55 105)(2 120 56 90)(3 81 49 107)(4 114 50 92)(5 83 51 109)(6 116 52 94)(7 85 53 111)(8 118 54 96)(9 36 80 101)(10 121 73 71)(11 38 74 103)(12 123 75 65)(13 40 76 97)(14 125 77 67)(15 34 78 99)(16 127 79 69)(17 119 31 89)(18 88 32 106)(19 113 25 91)(20 82 26 108)(21 115 27 93)(22 84 28 110)(23 117 29 95)(24 86 30 112)(33 61 98 45)(35 63 100 47)(37 57 102 41)(39 59 104 43)(42 122 58 72)(44 124 60 66)(46 126 62 68)(48 128 64 70)
(1 79 31 47)(2 76 32 44)(3 73 25 41)(4 78 26 46)(5 75 27 43)(6 80 28 48)(7 77 29 45)(8 74 30 42)(9 22 64 52)(10 19 57 49)(11 24 58 54)(12 21 59 51)(13 18 60 56)(14 23 61 53)(15 20 62 50)(16 17 63 55)(33 85 67 95)(34 82 68 92)(35 87 69 89)(36 84 70 94)(37 81 71 91)(38 86 72 96)(39 83 65 93)(40 88 66 90)(97 106 124 120)(98 111 125 117)(99 108 126 114)(100 105 127 119)(101 110 128 116)(102 107 121 113)(103 112 122 118)(104 109 123 115)
(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)| (1,87,55,105)(2,120,56,90)(3,81,49,107)(4,114,50,92)(5,83,51,109)(6,116,52,94)(7,85,53,111)(8,118,54,96)(9,36,80,101)(10,121,73,71)(11,38,74,103)(12,123,75,65)(13,40,76,97)(14,125,77,67)(15,34,78,99)(16,127,79,69)(17,119,31,89)(18,88,32,106)(19,113,25,91)(20,82,26,108)(21,115,27,93)(22,84,28,110)(23,117,29,95)(24,86,30,112)(33,61,98,45)(35,63,100,47)(37,57,102,41)(39,59,104,43)(42,122,58,72)(44,124,60,66)(46,126,62,68)(48,128,64,70), (1,79,31,47)(2,76,32,44)(3,73,25,41)(4,78,26,46)(5,75,27,43)(6,80,28,48)(7,77,29,45)(8,74,30,42)(9,22,64,52)(10,19,57,49)(11,24,58,54)(12,21,59,51)(13,18,60,56)(14,23,61,53)(15,20,62,50)(16,17,63,55)(33,85,67,95)(34,82,68,92)(35,87,69,89)(36,84,70,94)(37,81,71,91)(38,86,72,96)(39,83,65,93)(40,88,66,90)(97,106,124,120)(98,111,125,117)(99,108,126,114)(100,105,127,119)(101,110,128,116)(102,107,121,113)(103,112,122,118)(104,109,123,115), (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( (1,87,55,105)(2,120,56,90)(3,81,49,107)(4,114,50,92)(5,83,51,109)(6,116,52,94)(7,85,53,111)(8,118,54,96)(9,36,80,101)(10,121,73,71)(11,38,74,103)(12,123,75,65)(13,40,76,97)(14,125,77,67)(15,34,78,99)(16,127,79,69)(17,119,31,89)(18,88,32,106)(19,113,25,91)(20,82,26,108)(21,115,27,93)(22,84,28,110)(23,117,29,95)(24,86,30,112)(33,61,98,45)(35,63,100,47)(37,57,102,41)(39,59,104,43)(42,122,58,72)(44,124,60,66)(46,126,62,68)(48,128,64,70), (1,79,31,47)(2,76,32,44)(3,73,25,41)(4,78,26,46)(5,75,27,43)(6,80,28,48)(7,77,29,45)(8,74,30,42)(9,22,64,52)(10,19,57,49)(11,24,58,54)(12,21,59,51)(13,18,60,56)(14,23,61,53)(15,20,62,50)(16,17,63,55)(33,85,67,95)(34,82,68,92)(35,87,69,89)(36,84,70,94)(37,81,71,91)(38,86,72,96)(39,83,65,93)(40,88,66,90)(97,106,124,120)(98,111,125,117)(99,108,126,114)(100,105,127,119)(101,110,128,116)(102,107,121,113)(103,112,122,118)(104,109,123,115), (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([[(1,87,55,105),(2,120,56,90),(3,81,49,107),(4,114,50,92),(5,83,51,109),(6,116,52,94),(7,85,53,111),(8,118,54,96),(9,36,80,101),(10,121,73,71),(11,38,74,103),(12,123,75,65),(13,40,76,97),(14,125,77,67),(15,34,78,99),(16,127,79,69),(17,119,31,89),(18,88,32,106),(19,113,25,91),(20,82,26,108),(21,115,27,93),(22,84,28,110),(23,117,29,95),(24,86,30,112),(33,61,98,45),(35,63,100,47),(37,57,102,41),(39,59,104,43),(42,122,58,72),(44,124,60,66),(46,126,62,68),(48,128,64,70)], [(1,79,31,47),(2,76,32,44),(3,73,25,41),(4,78,26,46),(5,75,27,43),(6,80,28,48),(7,77,29,45),(8,74,30,42),(9,22,64,52),(10,19,57,49),(11,24,58,54),(12,21,59,51),(13,18,60,56),(14,23,61,53),(15,20,62,50),(16,17,63,55),(33,85,67,95),(34,82,68,92),(35,87,69,89),(36,84,70,94),(37,81,71,91),(38,86,72,96),(39,83,65,93),(40,88,66,90),(97,106,124,120),(98,111,125,117),(99,108,126,114),(100,105,127,119),(101,110,128,116),(102,107,121,113),(103,112,122,118),(104,109,123,115)], [(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 | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | M4(2) | C4○D4 |
| kernel | C43.7C2 | C22.7C42 | C43 | C2×C8⋊C4 | C2×C4⋊C8 | C4⋊C8 | C2×C42 | C42 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 1 | 2 | 2 | 16 | 8 | 2 | 2 | 16 | 4 |
Matrix representation of C43.7C2 ►in GL5(𝔽17)
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 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 | 5 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 15 |
| 0 | 0 | 0 | 15 | 14 |
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