p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.25Q8, C42.105D4, C4.45(C4⋊Q8), C4.45(C4⋊1D4), C42⋊4C4.8C2, C2.4(C42⋊9C4), C22.50(C8○D4), (C22×C8).24C22, C2.C42.18C4, (C2×C42).258C22, C23.311(C22×C4), (C22×C4).1626C23, C2.12(C42.6C22), (C2×C4⋊C8).27C2, (C2×C4).47(C4⋊C4), (C2×C4).340(C2×Q8), (C2×C4).1524(C2×D4), C22.101(C2×C4⋊C4), (C22×C4).270(C2×C4), SmallGroup(128,575)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.25Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2bc2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 188 in 128 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C2.C42, C4⋊C8, C2×C42, C22×C8, C42⋊4C4, C2×C4⋊C8, C42.25Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C8○D4, C42⋊9C4, C42.6C22, C42.25Q8
►Regular action on 128 points
Generators in S128
(1 65 47 77)(2 78 48 66)(3 67 41 79)(4 80 42 68)(5 69 43 73)(6 74 44 70)(7 71 45 75)(8 76 46 72)(9 100 122 20)(10 21 123 101)(11 102 124 22)(12 23 125 103)(13 104 126 24)(14 17 127 97)(15 98 128 18)(16 19 121 99)(25 95 105 63)(26 64 106 96)(27 89 107 57)(28 58 108 90)(29 91 109 59)(30 60 110 92)(31 93 111 61)(32 62 112 94)(33 51 113 83)(34 84 114 52)(35 53 115 85)(36 86 116 54)(37 55 117 87)(38 88 118 56)(39 49 119 81)(40 82 120 50)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 72 13 68)(10 65 14 69)(11 66 15 70)(12 67 16 71)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(73 123 77 127)(74 124 78 128)(75 125 79 121)(76 126 80 122)(81 107 85 111)(82 108 86 112)(83 109 87 105)(84 110 88 106)(89 115 93 119)(90 116 94 120)(91 117 95 113)(92 118 96 114)
(1 39 23 95)(2 96 24 40)(3 33 17 89)(4 90 18 34)(5 35 19 91)(6 92 20 36)(7 37 21 93)(8 94 22 38)(9 82 74 26)(10 27 75 83)(11 84 76 28)(12 29 77 85)(13 86 78 30)(14 31 79 87)(15 88 80 32)(16 25 73 81)(41 113 97 57)(42 58 98 114)(43 115 99 59)(44 60 100 116)(45 117 101 61)(46 62 102 118)(47 119 103 63)(48 64 104 120)(49 121 105 69)(50 70 106 122)(51 123 107 71)(52 72 108 124)(53 125 109 65)(54 66 110 126)(55 127 111 67)(56 68 112 128)
(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)| (1,65,47,77)(2,78,48,66)(3,67,41,79)(4,80,42,68)(5,69,43,73)(6,74,44,70)(7,71,45,75)(8,76,46,72)(9,100,122,20)(10,21,123,101)(11,102,124,22)(12,23,125,103)(13,104,126,24)(14,17,127,97)(15,98,128,18)(16,19,121,99)(25,95,105,63)(26,64,106,96)(27,89,107,57)(28,58,108,90)(29,91,109,59)(30,60,110,92)(31,93,111,61)(32,62,112,94)(33,51,113,83)(34,84,114,52)(35,53,115,85)(36,86,116,54)(37,55,117,87)(38,88,118,56)(39,49,119,81)(40,82,120,50), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,115,93,119)(90,116,94,120)(91,117,95,113)(92,118,96,114), (1,39,23,95)(2,96,24,40)(3,33,17,89)(4,90,18,34)(5,35,19,91)(6,92,20,36)(7,37,21,93)(8,94,22,38)(9,82,74,26)(10,27,75,83)(11,84,76,28)(12,29,77,85)(13,86,78,30)(14,31,79,87)(15,88,80,32)(16,25,73,81)(41,113,97,57)(42,58,98,114)(43,115,99,59)(44,60,100,116)(45,117,101,61)(46,62,102,118)(47,119,103,63)(48,64,104,120)(49,121,105,69)(50,70,106,122)(51,123,107,71)(52,72,108,124)(53,125,109,65)(54,66,110,126)(55,127,111,67)(56,68,112,128), (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( (1,65,47,77)(2,78,48,66)(3,67,41,79)(4,80,42,68)(5,69,43,73)(6,74,44,70)(7,71,45,75)(8,76,46,72)(9,100,122,20)(10,21,123,101)(11,102,124,22)(12,23,125,103)(13,104,126,24)(14,17,127,97)(15,98,128,18)(16,19,121,99)(25,95,105,63)(26,64,106,96)(27,89,107,57)(28,58,108,90)(29,91,109,59)(30,60,110,92)(31,93,111,61)(32,62,112,94)(33,51,113,83)(34,84,114,52)(35,53,115,85)(36,86,116,54)(37,55,117,87)(38,88,118,56)(39,49,119,81)(40,82,120,50), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,115,93,119)(90,116,94,120)(91,117,95,113)(92,118,96,114), (1,39,23,95)(2,96,24,40)(3,33,17,89)(4,90,18,34)(5,35,19,91)(6,92,20,36)(7,37,21,93)(8,94,22,38)(9,82,74,26)(10,27,75,83)(11,84,76,28)(12,29,77,85)(13,86,78,30)(14,31,79,87)(15,88,80,32)(16,25,73,81)(41,113,97,57)(42,58,98,114)(43,115,99,59)(44,60,100,116)(45,117,101,61)(46,62,102,118)(47,119,103,63)(48,64,104,120)(49,121,105,69)(50,70,106,122)(51,123,107,71)(52,72,108,124)(53,125,109,65)(54,66,110,126)(55,127,111,67)(56,68,112,128), (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([[(1,65,47,77),(2,78,48,66),(3,67,41,79),(4,80,42,68),(5,69,43,73),(6,74,44,70),(7,71,45,75),(8,76,46,72),(9,100,122,20),(10,21,123,101),(11,102,124,22),(12,23,125,103),(13,104,126,24),(14,17,127,97),(15,98,128,18),(16,19,121,99),(25,95,105,63),(26,64,106,96),(27,89,107,57),(28,58,108,90),(29,91,109,59),(30,60,110,92),(31,93,111,61),(32,62,112,94),(33,51,113,83),(34,84,114,52),(35,53,115,85),(36,86,116,54),(37,55,117,87),(38,88,118,56),(39,49,119,81),(40,82,120,50)], [(1,97,5,101),(2,98,6,102),(3,99,7,103),(4,100,8,104),(9,72,13,68),(10,65,14,69),(11,66,15,70),(12,67,16,71),(17,43,21,47),(18,44,22,48),(19,45,23,41),(20,46,24,42),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58),(73,123,77,127),(74,124,78,128),(75,125,79,121),(76,126,80,122),(81,107,85,111),(82,108,86,112),(83,109,87,105),(84,110,88,106),(89,115,93,119),(90,116,94,120),(91,117,95,113),(92,118,96,114)], [(1,39,23,95),(2,96,24,40),(3,33,17,89),(4,90,18,34),(5,35,19,91),(6,92,20,36),(7,37,21,93),(8,94,22,38),(9,82,74,26),(10,27,75,83),(11,84,76,28),(12,29,77,85),(13,86,78,30),(14,31,79,87),(15,88,80,32),(16,25,73,81),(41,113,97,57),(42,58,98,114),(43,115,99,59),(44,60,100,116),(45,117,101,61),(46,62,102,118),(47,119,103,63),(48,64,104,120),(49,121,105,69),(50,70,106,122),(51,123,107,71),(52,72,108,124),(53,125,109,65),(54,66,110,126),(55,127,111,67),(56,68,112,128)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | C8○D4 |
| kernel | C42.25Q8 | C42⋊4C4 | C2×C4⋊C8 | C2.C42 | C42 | C42 | C22 |
| # reps | 1 | 1 | 6 | 8 | 6 | 6 | 16 |
Matrix representation of C42.25Q8 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 4 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 0 | 9 | 7 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,11,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,15,0,0,0,0,15,0,0,0,0,0,0,0,1,4,0,0,0,0,8,16,0,0,0,0,0,0,10,9,0,0,0,0,2,7] >;
C42.25Q8 in GAP, Magma, Sage, TeX
C_4^2._{25}Q_8 % in TeX
G:=Group("C4^2.25Q8"); // GroupNames label
G:=SmallGroup(128,575);
// by ID
G=gap.SmallGroup(128,575);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b*c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations