p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.45Q8, C42.381D4, C4⋊C8⋊12C4, C4.43(C4×Q8), C4.167(C4×D4), (C2×C4).25C42, C42.131(C2×C4), C42⋊4C4.6C2, C22.25(C8○D4), C22.54(C2×C42), C2.C42.15C4, (C22×C8).476C22, C23.258(C22×C4), (C2×C42).238C22, C2.15(C8○2M4(2)), (C22×C4).1611C23, C22.53(C42⋊C2), C22.7C42.41C2, C2.3(C42.7C22), C2.2(C42.6C22), C2.8(C4×C4⋊C4), (C2×C4×C8).15C2, (C2×C4⋊C8).52C2, (C2×C4).79(C4⋊C4), (C2×C8).136(C2×C4), C22.57(C2×C4⋊C4), (C2×C4).331(C2×Q8), (C2×C8⋊C4).25C2, (C2×C4).1503(C2×D4), (C2×C4).921(C4○D4), (C22×C4).109(C2×C4), (C2×C4).601(C22×C4), SmallGroup(128,500)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.45Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 180 in 130 conjugacy classes, 84 normal (16 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, C2.C42, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C22×C8, C22.7C42, C42⋊4C4, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8, C42.45Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C8○D4, C4×C4⋊C4, C8○2M4(2), C42.6C22, C42.7C22, C42.45Q8
►Regular action on 128 points
Generators in S128
(1 79 31 12)(2 80 32 13)(3 73 25 14)(4 74 26 15)(5 75 27 16)(6 76 28 9)(7 77 29 10)(8 78 30 11)(17 57 49 41)(18 58 50 42)(19 59 51 43)(20 60 52 44)(21 61 53 45)(22 62 54 46)(23 63 55 47)(24 64 56 48)(33 81 65 89)(34 82 66 90)(35 83 67 91)(36 84 68 92)(37 85 69 93)(38 86 70 94)(39 87 71 95)(40 88 72 96)(97 106 126 114)(98 107 127 115)(99 108 128 116)(100 109 121 117)(101 110 122 118)(102 111 123 119)(103 112 124 120)(104 105 125 113)
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 62 13 58)(10 63 14 59)(11 64 15 60)(12 57 16 61)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)(33 98 37 102)(34 99 38 103)(35 100 39 104)(36 101 40 97)(41 75 45 79)(42 76 46 80)(43 77 47 73)(44 78 48 74)(65 127 69 123)(66 128 70 124)(67 121 71 125)(68 122 72 126)(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 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 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)
G:=sub<Sym(128)| (1,79,31,12)(2,80,32,13)(3,73,25,14)(4,74,26,15)(5,75,27,16)(6,76,28,9)(7,77,29,10)(8,78,30,11)(17,57,49,41)(18,58,50,42)(19,59,51,43)(20,60,52,44)(21,61,53,45)(22,62,54,46)(23,63,55,47)(24,64,56,48)(33,81,65,89)(34,82,66,90)(35,83,67,91)(36,84,68,92)(37,85,69,93)(38,86,70,94)(39,87,71,95)(40,88,72,96)(97,106,126,114)(98,107,127,115)(99,108,128,116)(100,109,121,117)(101,110,122,118)(102,111,123,119)(103,112,124,120)(104,105,125,113), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,62,13,58)(10,63,14,59)(11,64,15,60)(12,57,16,61)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,98,37,102)(34,99,38,103)(35,100,39,104)(36,101,40,97)(41,75,45,79)(42,76,46,80)(43,77,47,73)(44,78,48,74)(65,127,69,123)(66,128,70,124)(67,121,71,125)(68,122,72,126)(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), (1,2,3,4,5,6,7,8)(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), (1,87,55,111)(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)>;
G:=Group( (1,79,31,12)(2,80,32,13)(3,73,25,14)(4,74,26,15)(5,75,27,16)(6,76,28,9)(7,77,29,10)(8,78,30,11)(17,57,49,41)(18,58,50,42)(19,59,51,43)(20,60,52,44)(21,61,53,45)(22,62,54,46)(23,63,55,47)(24,64,56,48)(33,81,65,89)(34,82,66,90)(35,83,67,91)(36,84,68,92)(37,85,69,93)(38,86,70,94)(39,87,71,95)(40,88,72,96)(97,106,126,114)(98,107,127,115)(99,108,128,116)(100,109,121,117)(101,110,122,118)(102,111,123,119)(103,112,124,120)(104,105,125,113), (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,62,13,58)(10,63,14,59)(11,64,15,60)(12,57,16,61)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,98,37,102)(34,99,38,103)(35,100,39,104)(36,101,40,97)(41,75,45,79)(42,76,46,80)(43,77,47,73)(44,78,48,74)(65,127,69,123)(66,128,70,124)(67,121,71,125)(68,122,72,126)(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), (1,2,3,4,5,6,7,8)(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), (1,87,55,111)(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) );
G=PermutationGroup([[(1,79,31,12),(2,80,32,13),(3,73,25,14),(4,74,26,15),(5,75,27,16),(6,76,28,9),(7,77,29,10),(8,78,30,11),(17,57,49,41),(18,58,50,42),(19,59,51,43),(20,60,52,44),(21,61,53,45),(22,62,54,46),(23,63,55,47),(24,64,56,48),(33,81,65,89),(34,82,66,90),(35,83,67,91),(36,84,68,92),(37,85,69,93),(38,86,70,94),(39,87,71,95),(40,88,72,96),(97,106,126,114),(98,107,127,115),(99,108,128,116),(100,109,121,117),(101,110,122,118),(102,111,123,119),(103,112,124,120),(104,105,125,113)], [(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,62,13,58),(10,63,14,59),(11,64,15,60),(12,57,16,61),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26),(33,98,37,102),(34,99,38,103),(35,100,39,104),(36,101,40,97),(41,75,45,79),(42,76,46,80),(43,77,47,73),(44,78,48,74),(65,127,69,123),(66,128,70,124),(67,121,71,125),(68,122,72,126),(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)], [(1,2,3,4,5,6,7,8),(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)], [(1,87,55,111),(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 | C8○D4 |
| kernel | C42.45Q8 | C22.7C42 | C42⋊4C4 | C2×C4×C8 | C2×C8⋊C4 | C2×C4⋊C8 | C2.C42 | C4⋊C8 | C42 | C42 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 16 | 2 | 2 | 4 | 16 |
Matrix representation of C42.45Q8 ►in GL5(𝔽17)
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 0 | 16 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 16 |
| 0 | 0 | 0 | 8 | 15 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 11 | 0 | 0 |
| 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 9 | 9 |
| 0 | 0 | 0 | 6 | 8 |
G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,16,0,0,0,15,13],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,2,8,0,0,0,16,15],[13,0,0,0,0,0,4,11,0,0,0,11,13,0,0,0,0,0,9,6,0,0,0,9,8] >;
C42.45Q8 in GAP, Magma, Sage, TeX
C_4^2._{45}Q_8 % in TeX
G:=Group("C4^2.45Q8"); // GroupNames label
G:=SmallGroup(128,500);
// by ID
G=gap.SmallGroup(128,500);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations