p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.4Q8, (C2×C8).166D4, C4.37(C4⋊Q8), (C2×C4).42SD16, C2.23(C8⋊8D4), C2.23(C8⋊D4), C2.9(Q8⋊Q8), C2.9(D4⋊2Q8), (C22×C4).323D4, C23.938(C2×D4), C2.13(D4.Q8), C2.13(Q8.Q8), C4.24(C42.C2), C22.128(C4○D8), C22.4Q16.29C2, (C22×C8).328C22, (C2×C42).389C22, C22.107(C2×SD16), C22.261(C4⋊D4), C22.157(C8⋊C22), (C22×C4).1472C23, C22.115(C22⋊Q8), C4.117(C22.D4), C22.146(C8.C22), C23.65C23.25C2, C2.10(C23.81C23), (C2×C4⋊C8).49C2, (C2×C4).289(C2×Q8), (C2×C4.Q8).27C2, (C2×C4).1381(C2×D4), (C2×C4).631(C4○D4), (C2×C4⋊C4).159C22, SmallGroup(128,820)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.(C4⋊Q8)
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a-1c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a-1b-1, dcd-1=c3 >
Subgroups: 224 in 112 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4⋊C8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.4Q16, C23.65C23, C2×C4⋊C8, C2×C4.Q8, C4.(C4⋊Q8)
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.81C23, C8⋊8D4, C8⋊D4, Q8⋊Q8, D4⋊2Q8, D4.Q8, Q8.Q8, C4.(C4⋊Q8)
►Regular action on 128 points
Generators in S128
(1 85 5 81)(2 86 6 82)(3 87 7 83)(4 88 8 84)(9 74 13 78)(10 75 14 79)(11 76 15 80)(12 77 16 73)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 95 29 91)(26 96 30 92)(27 89 31 93)(28 90 32 94)(41 99 45 103)(42 100 46 104)(43 101 47 97)(44 102 48 98)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 115 61 119)(58 116 62 120)(59 117 63 113)(60 118 64 114)(65 126 69 122)(66 127 70 123)(67 128 71 124)(68 121 72 125)
(1 122 47 73)(2 66 48 13)(3 128 41 79)(4 72 42 11)(5 126 43 77)(6 70 44 9)(7 124 45 75)(8 68 46 15)(10 83 71 103)(12 81 65 101)(14 87 67 99)(16 85 69 97)(17 28 117 56)(18 91 118 107)(19 26 119 54)(20 89 120 105)(21 32 113 52)(22 95 114 111)(23 30 115 50)(24 93 116 109)(25 60 53 36)(27 58 55 34)(29 64 49 40)(31 62 51 38)(33 92 57 108)(35 90 59 106)(37 96 61 112)(39 94 63 110)(74 86 123 98)(76 84 125 104)(78 82 127 102)(80 88 121 100)
(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 91 83 31)(2 94 84 26)(3 89 85 29)(4 92 86 32)(5 95 87 27)(6 90 88 30)(7 93 81 25)(8 96 82 28)(9 117 80 61)(10 120 73 64)(11 115 74 59)(12 118 75 62)(13 113 76 57)(14 116 77 60)(15 119 78 63)(16 114 79 58)(17 121 37 70)(18 124 38 65)(19 127 39 68)(20 122 40 71)(21 125 33 66)(22 128 34 69)(23 123 35 72)(24 126 36 67)(41 105 97 49)(42 108 98 52)(43 111 99 55)(44 106 100 50)(45 109 101 53)(46 112 102 56)(47 107 103 51)(48 110 104 54)
G:=sub<Sym(128)| (1,85,5,81)(2,86,6,82)(3,87,7,83)(4,88,8,84)(9,74,13,78)(10,75,14,79)(11,76,15,80)(12,77,16,73)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,95,29,91)(26,96,30,92)(27,89,31,93)(28,90,32,94)(41,99,45,103)(42,100,46,104)(43,101,47,97)(44,102,48,98)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,115,61,119)(58,116,62,120)(59,117,63,113)(60,118,64,114)(65,126,69,122)(66,127,70,123)(67,128,71,124)(68,121,72,125), (1,122,47,73)(2,66,48,13)(3,128,41,79)(4,72,42,11)(5,126,43,77)(6,70,44,9)(7,124,45,75)(8,68,46,15)(10,83,71,103)(12,81,65,101)(14,87,67,99)(16,85,69,97)(17,28,117,56)(18,91,118,107)(19,26,119,54)(20,89,120,105)(21,32,113,52)(22,95,114,111)(23,30,115,50)(24,93,116,109)(25,60,53,36)(27,58,55,34)(29,64,49,40)(31,62,51,38)(33,92,57,108)(35,90,59,106)(37,96,61,112)(39,94,63,110)(74,86,123,98)(76,84,125,104)(78,82,127,102)(80,88,121,100), (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), (1,91,83,31)(2,94,84,26)(3,89,85,29)(4,92,86,32)(5,95,87,27)(6,90,88,30)(7,93,81,25)(8,96,82,28)(9,117,80,61)(10,120,73,64)(11,115,74,59)(12,118,75,62)(13,113,76,57)(14,116,77,60)(15,119,78,63)(16,114,79,58)(17,121,37,70)(18,124,38,65)(19,127,39,68)(20,122,40,71)(21,125,33,66)(22,128,34,69)(23,123,35,72)(24,126,36,67)(41,105,97,49)(42,108,98,52)(43,111,99,55)(44,106,100,50)(45,109,101,53)(46,112,102,56)(47,107,103,51)(48,110,104,54)>;
G:=Group( (1,85,5,81)(2,86,6,82)(3,87,7,83)(4,88,8,84)(9,74,13,78)(10,75,14,79)(11,76,15,80)(12,77,16,73)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,95,29,91)(26,96,30,92)(27,89,31,93)(28,90,32,94)(41,99,45,103)(42,100,46,104)(43,101,47,97)(44,102,48,98)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,115,61,119)(58,116,62,120)(59,117,63,113)(60,118,64,114)(65,126,69,122)(66,127,70,123)(67,128,71,124)(68,121,72,125), (1,122,47,73)(2,66,48,13)(3,128,41,79)(4,72,42,11)(5,126,43,77)(6,70,44,9)(7,124,45,75)(8,68,46,15)(10,83,71,103)(12,81,65,101)(14,87,67,99)(16,85,69,97)(17,28,117,56)(18,91,118,107)(19,26,119,54)(20,89,120,105)(21,32,113,52)(22,95,114,111)(23,30,115,50)(24,93,116,109)(25,60,53,36)(27,58,55,34)(29,64,49,40)(31,62,51,38)(33,92,57,108)(35,90,59,106)(37,96,61,112)(39,94,63,110)(74,86,123,98)(76,84,125,104)(78,82,127,102)(80,88,121,100), (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), (1,91,83,31)(2,94,84,26)(3,89,85,29)(4,92,86,32)(5,95,87,27)(6,90,88,30)(7,93,81,25)(8,96,82,28)(9,117,80,61)(10,120,73,64)(11,115,74,59)(12,118,75,62)(13,113,76,57)(14,116,77,60)(15,119,78,63)(16,114,79,58)(17,121,37,70)(18,124,38,65)(19,127,39,68)(20,122,40,71)(21,125,33,66)(22,128,34,69)(23,123,35,72)(24,126,36,67)(41,105,97,49)(42,108,98,52)(43,111,99,55)(44,106,100,50)(45,109,101,53)(46,112,102,56)(47,107,103,51)(48,110,104,54) );
G=PermutationGroup([[(1,85,5,81),(2,86,6,82),(3,87,7,83),(4,88,8,84),(9,74,13,78),(10,75,14,79),(11,76,15,80),(12,77,16,73),(17,39,21,35),(18,40,22,36),(19,33,23,37),(20,34,24,38),(25,95,29,91),(26,96,30,92),(27,89,31,93),(28,90,32,94),(41,99,45,103),(42,100,46,104),(43,101,47,97),(44,102,48,98),(49,107,53,111),(50,108,54,112),(51,109,55,105),(52,110,56,106),(57,115,61,119),(58,116,62,120),(59,117,63,113),(60,118,64,114),(65,126,69,122),(66,127,70,123),(67,128,71,124),(68,121,72,125)], [(1,122,47,73),(2,66,48,13),(3,128,41,79),(4,72,42,11),(5,126,43,77),(6,70,44,9),(7,124,45,75),(8,68,46,15),(10,83,71,103),(12,81,65,101),(14,87,67,99),(16,85,69,97),(17,28,117,56),(18,91,118,107),(19,26,119,54),(20,89,120,105),(21,32,113,52),(22,95,114,111),(23,30,115,50),(24,93,116,109),(25,60,53,36),(27,58,55,34),(29,64,49,40),(31,62,51,38),(33,92,57,108),(35,90,59,106),(37,96,61,112),(39,94,63,110),(74,86,123,98),(76,84,125,104),(78,82,127,102),(80,88,121,100)], [(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)], [(1,91,83,31),(2,94,84,26),(3,89,85,29),(4,92,86,32),(5,95,87,27),(6,90,88,30),(7,93,81,25),(8,96,82,28),(9,117,80,61),(10,120,73,64),(11,115,74,59),(12,118,75,62),(13,113,76,57),(14,116,77,60),(15,119,78,63),(16,114,79,58),(17,121,37,70),(18,124,38,65),(19,127,39,68),(20,122,40,71),(21,125,33,66),(22,128,34,69),(23,123,35,72),(24,126,36,67),(41,105,97,49),(42,108,98,52),(43,111,99,55),(44,106,100,50),(45,109,101,53),(46,112,102,56),(47,107,103,51),(48,110,104,54)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D4 | SD16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C4.(C4⋊Q8) | C22.4Q16 | C23.65C23 | C2×C4⋊C8 | C2×C4.Q8 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 6 | 4 | 1 | 1 |
Matrix representation of C4.(C4⋊Q8) ►in GL6(𝔽17)
| 1 | 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 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,0,8,0,0,0,0,2,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,8,0,0,0,0,0,0,2],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C4.(C4⋊Q8) in GAP, Magma, Sage, TeX
C_4.(C_4\rtimes Q_8)
% in TeX
G:=Group("C4.(C4:Q8)"); // GroupNames label
G:=SmallGroup(128,820);
// by ID
G=gap.SmallGroup(128,820);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^-1*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^3>;
// generators/relations