p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.191D4, C23.534C24, C22.2282- 1+4, C42⋊5C4.12C2, (C2×C42).611C22, (C22×C4).144C23, C22.359(C22×D4), (C22×Q8).450C22, C2.85(C22.19C24), C23.83C23.23C2, C2.C42.259C22, C23.78C23.15C2, C23.81C23.26C2, C23.63C23.37C2, C2.27(C22.35C24), C2.42(C23.38C23), (C2×C4×Q8).40C2, (C2×C4).393(C2×D4), (C2×C4).169(C4○D4), (C2×C4⋊C4).361C22, C22.406(C2×C4○D4), (C2×C42.C2).23C2, SmallGroup(128,1366)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.191D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c-1 >
Subgroups: 340 in 208 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C42.C2, C22×Q8, C42⋊5C4, C23.63C23, C23.78C23, C23.81C23, C23.83C23, C2×C4×Q8, C2×C42.C2, C42.191D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C22.19C24, C23.38C23, C22.35C24, C42.191D4
►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 13 107 44)(2 14 108 41)(3 15 105 42)(4 16 106 43)(5 66 93 40)(6 67 94 37)(7 68 95 38)(8 65 96 39)(9 103 48 76)(10 104 45 73)(11 101 46 74)(12 102 47 75)(17 109 56 82)(18 110 53 83)(19 111 54 84)(20 112 55 81)(21 113 52 78)(22 114 49 79)(23 115 50 80)(24 116 51 77)(25 119 64 92)(26 120 61 89)(27 117 62 90)(28 118 63 91)(29 123 60 88)(30 124 57 85)(31 121 58 86)(32 122 59 87)(33 125 71 98)(34 126 72 99)(35 127 69 100)(36 128 70 97)
(1 125 101 96)(2 97 102 7)(3 127 103 94)(4 99 104 5)(6 105 100 76)(8 107 98 74)(9 65 42 33)(10 38 43 70)(11 67 44 35)(12 40 41 72)(13 69 46 37)(14 34 47 66)(15 71 48 39)(16 36 45 68)(17 60 50 27)(18 32 51 61)(19 58 52 25)(20 30 49 63)(21 64 54 31)(22 28 55 57)(23 62 56 29)(24 26 53 59)(73 93 106 126)(75 95 108 128)(77 91 110 124)(78 117 111 88)(79 89 112 122)(80 119 109 86)(81 87 114 120)(82 121 115 92)(83 85 116 118)(84 123 113 90)
(1 86 3 88)(2 85 4 87)(5 114 7 116)(6 113 8 115)(9 27 11 25)(10 26 12 28)(13 31 15 29)(14 30 16 32)(17 35 19 33)(18 34 20 36)(21 39 23 37)(22 38 24 40)(41 57 43 59)(42 60 44 58)(45 61 47 63)(46 64 48 62)(49 68 51 66)(50 67 52 65)(53 72 55 70)(54 71 56 69)(73 89 75 91)(74 92 76 90)(77 93 79 95)(78 96 80 94)(81 97 83 99)(82 100 84 98)(101 119 103 117)(102 118 104 120)(105 123 107 121)(106 122 108 124)(109 127 111 125)(110 126 112 128)
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), (1,13,107,44)(2,14,108,41)(3,15,105,42)(4,16,106,43)(5,66,93,40)(6,67,94,37)(7,68,95,38)(8,65,96,39)(9,103,48,76)(10,104,45,73)(11,101,46,74)(12,102,47,75)(17,109,56,82)(18,110,53,83)(19,111,54,84)(20,112,55,81)(21,113,52,78)(22,114,49,79)(23,115,50,80)(24,116,51,77)(25,119,64,92)(26,120,61,89)(27,117,62,90)(28,118,63,91)(29,123,60,88)(30,124,57,85)(31,121,58,86)(32,122,59,87)(33,125,71,98)(34,126,72,99)(35,127,69,100)(36,128,70,97), (1,125,101,96)(2,97,102,7)(3,127,103,94)(4,99,104,5)(6,105,100,76)(8,107,98,74)(9,65,42,33)(10,38,43,70)(11,67,44,35)(12,40,41,72)(13,69,46,37)(14,34,47,66)(15,71,48,39)(16,36,45,68)(17,60,50,27)(18,32,51,61)(19,58,52,25)(20,30,49,63)(21,64,54,31)(22,28,55,57)(23,62,56,29)(24,26,53,59)(73,93,106,126)(75,95,108,128)(77,91,110,124)(78,117,111,88)(79,89,112,122)(80,119,109,86)(81,87,114,120)(82,121,115,92)(83,85,116,118)(84,123,113,90), (1,86,3,88)(2,85,4,87)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,57,43,59)(42,60,44,58)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,89,75,91)(74,92,76,90)(77,93,79,95)(78,96,80,94)(81,97,83,99)(82,100,84,98)(101,119,103,117)(102,118,104,120)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128)>;
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), (1,13,107,44)(2,14,108,41)(3,15,105,42)(4,16,106,43)(5,66,93,40)(6,67,94,37)(7,68,95,38)(8,65,96,39)(9,103,48,76)(10,104,45,73)(11,101,46,74)(12,102,47,75)(17,109,56,82)(18,110,53,83)(19,111,54,84)(20,112,55,81)(21,113,52,78)(22,114,49,79)(23,115,50,80)(24,116,51,77)(25,119,64,92)(26,120,61,89)(27,117,62,90)(28,118,63,91)(29,123,60,88)(30,124,57,85)(31,121,58,86)(32,122,59,87)(33,125,71,98)(34,126,72,99)(35,127,69,100)(36,128,70,97), (1,125,101,96)(2,97,102,7)(3,127,103,94)(4,99,104,5)(6,105,100,76)(8,107,98,74)(9,65,42,33)(10,38,43,70)(11,67,44,35)(12,40,41,72)(13,69,46,37)(14,34,47,66)(15,71,48,39)(16,36,45,68)(17,60,50,27)(18,32,51,61)(19,58,52,25)(20,30,49,63)(21,64,54,31)(22,28,55,57)(23,62,56,29)(24,26,53,59)(73,93,106,126)(75,95,108,128)(77,91,110,124)(78,117,111,88)(79,89,112,122)(80,119,109,86)(81,87,114,120)(82,121,115,92)(83,85,116,118)(84,123,113,90), (1,86,3,88)(2,85,4,87)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,57,43,59)(42,60,44,58)(45,61,47,63)(46,64,48,62)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,89,75,91)(74,92,76,90)(77,93,79,95)(78,96,80,94)(81,97,83,99)(82,100,84,98)(101,119,103,117)(102,118,104,120)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128) );
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)], [(1,13,107,44),(2,14,108,41),(3,15,105,42),(4,16,106,43),(5,66,93,40),(6,67,94,37),(7,68,95,38),(8,65,96,39),(9,103,48,76),(10,104,45,73),(11,101,46,74),(12,102,47,75),(17,109,56,82),(18,110,53,83),(19,111,54,84),(20,112,55,81),(21,113,52,78),(22,114,49,79),(23,115,50,80),(24,116,51,77),(25,119,64,92),(26,120,61,89),(27,117,62,90),(28,118,63,91),(29,123,60,88),(30,124,57,85),(31,121,58,86),(32,122,59,87),(33,125,71,98),(34,126,72,99),(35,127,69,100),(36,128,70,97)], [(1,125,101,96),(2,97,102,7),(3,127,103,94),(4,99,104,5),(6,105,100,76),(8,107,98,74),(9,65,42,33),(10,38,43,70),(11,67,44,35),(12,40,41,72),(13,69,46,37),(14,34,47,66),(15,71,48,39),(16,36,45,68),(17,60,50,27),(18,32,51,61),(19,58,52,25),(20,30,49,63),(21,64,54,31),(22,28,55,57),(23,62,56,29),(24,26,53,59),(73,93,106,126),(75,95,108,128),(77,91,110,124),(78,117,111,88),(79,89,112,122),(80,119,109,86),(81,87,114,120),(82,121,115,92),(83,85,116,118),(84,123,113,90)], [(1,86,3,88),(2,85,4,87),(5,114,7,116),(6,113,8,115),(9,27,11,25),(10,26,12,28),(13,31,15,29),(14,30,16,32),(17,35,19,33),(18,34,20,36),(21,39,23,37),(22,38,24,40),(41,57,43,59),(42,60,44,58),(45,61,47,63),(46,64,48,62),(49,68,51,66),(50,67,52,65),(53,72,55,70),(54,71,56,69),(73,89,75,91),(74,92,76,90),(77,93,79,95),(78,96,80,94),(81,97,83,99),(82,100,84,98),(101,119,103,117),(102,118,104,120),(105,123,107,121),(106,122,108,124),(109,127,111,125),(110,126,112,128)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.191D4 | C42⋊5C4 | C23.63C23 | C23.78C23 | C23.81C23 | C23.83C23 | C2×C4×Q8 | C2×C42.C2 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 4 | 8 | 4 |
Matrix representation of C42.191D4 ►in GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 3 |
| 0 | 0 | 0 | 0 | 2 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 2 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 | 3 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 1 |
| 0 | 0 | 0 | 0 | 4 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,2,1,0,0,0,0,0,3,1,3,0,0,0,0,2,1,2,0,0,0,0,0,1,3,0,2],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,2,4,0,0,0,0,0,1,1,0,4,0,0,0,0,4,0,1,3,0,0,0,0,0,4,4,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,4,2,0,0,0,0,0,1,1,1,0,0,0,0,4,2,4,0,0,0,0,0,1,1,0,4] >;
C42.191D4 in GAP, Magma, Sage, TeX
C_4^2._{191}D_4 % in TeX
G:=Group("C4^2.191D4"); // GroupNames label
G:=SmallGroup(128,1366);
// by ID
G=gap.SmallGroup(128,1366);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations