p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.1Q8, C2.8(C8⋊Q8), (C2×C8).22Q8, C4⋊C4.101D4, C4.23(C4⋊Q8), C2.9(Q8.Q8), C2.9(D4.Q8), C4.8(C22⋊Q8), (C22×C4).153D4, C23.921(C2×D4), C22.50(C4⋊Q8), C2.5(C8.5Q8), C2.34(D4⋊D4), C22.229C22≀C2, C2.34(D4.7D4), C22.115(C4○D8), C22.4Q16.39C2, (C22×C8).324C22, (C2×C42).373C22, C22.144(C8⋊C22), (C22×C4).1455C23, C22.108(C22⋊Q8), C22.133(C8.C22), C22.7C42.27C2, C23.65C23.17C2, C2.6(C23.78C23), (C2×C4).219(C2×Q8), (C2×C2.D8).13C2, (C2×C4.Q8).23C2, (C2×C4).1050(C2×D4), (C2×C4).777(C4○D4), (C2×C4⋊C4).136C22, (C2×C42.C2).10C2, SmallGroup(128,791)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.(C8⋊Q8)
G = < a,b,c,d | a2=b8=c4=1, d2=ac2, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=ab3, dcd-1=ac-1 >
Subgroups: 240 in 125 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C22×C8, C22.7C42, C22.4Q16, C23.65C23, C2×C4.Q8, C2×C2.D8, C2×C42.C2, C2.(C8⋊Q8)
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4⋊Q8, C4○D8, C8⋊C22, C8.C22, C23.78C23, D4⋊D4, D4.7D4, D4.Q8, Q8.Q8, C8.5Q8, C8⋊Q8, C2.(C8⋊Q8)
►Regular action on 128 points
Generators in S128
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 84)(34 85)(35 86)(36 87)(37 88)(38 81)(39 82)(40 83)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 113)(56 114)(73 124)(74 125)(75 126)(76 127)(77 128)(78 121)(79 122)(80 123)
(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)
(1 127 65 96)(2 73 66 13)(3 121 67 90)(4 75 68 15)(5 123 69 92)(6 77 70 9)(7 125 71 94)(8 79 72 11)(10 107 78 45)(12 109 80 47)(14 111 74 41)(16 105 76 43)(17 81 28 50)(18 35 29 113)(19 83 30 52)(20 37 31 115)(21 85 32 54)(22 39 25 117)(23 87 26 56)(24 33 27 119)(34 102 120 63)(36 104 114 57)(38 98 116 59)(40 100 118 61)(42 91 112 122)(44 93 106 124)(46 95 108 126)(48 89 110 128)(49 62 88 101)(51 64 82 103)(53 58 84 97)(55 60 86 99)
(1 49 43 37)(2 118 44 83)(3 55 45 35)(4 116 46 81)(5 53 47 33)(6 114 48 87)(7 51 41 39)(8 120 42 85)(9 23 128 104)(10 60 121 29)(11 21 122 102)(12 58 123 27)(13 19 124 100)(14 64 125 25)(15 17 126 98)(16 62 127 31)(18 78 99 90)(20 76 101 96)(22 74 103 94)(24 80 97 92)(26 89 57 77)(28 95 59 75)(30 93 61 73)(32 91 63 79)(34 112 54 72)(36 110 56 70)(38 108 50 68)(40 106 52 66)(65 88 105 115)(67 86 107 113)(69 84 109 119)(71 82 111 117)
G:=sub<Sym(128)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,84)(34,85)(35,86)(36,87)(37,88)(38,81)(39,82)(40,83)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,113)(56,114)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123), (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), (1,127,65,96)(2,73,66,13)(3,121,67,90)(4,75,68,15)(5,123,69,92)(6,77,70,9)(7,125,71,94)(8,79,72,11)(10,107,78,45)(12,109,80,47)(14,111,74,41)(16,105,76,43)(17,81,28,50)(18,35,29,113)(19,83,30,52)(20,37,31,115)(21,85,32,54)(22,39,25,117)(23,87,26,56)(24,33,27,119)(34,102,120,63)(36,104,114,57)(38,98,116,59)(40,100,118,61)(42,91,112,122)(44,93,106,124)(46,95,108,126)(48,89,110,128)(49,62,88,101)(51,64,82,103)(53,58,84,97)(55,60,86,99), (1,49,43,37)(2,118,44,83)(3,55,45,35)(4,116,46,81)(5,53,47,33)(6,114,48,87)(7,51,41,39)(8,120,42,85)(9,23,128,104)(10,60,121,29)(11,21,122,102)(12,58,123,27)(13,19,124,100)(14,64,125,25)(15,17,126,98)(16,62,127,31)(18,78,99,90)(20,76,101,96)(22,74,103,94)(24,80,97,92)(26,89,57,77)(28,95,59,75)(30,93,61,73)(32,91,63,79)(34,112,54,72)(36,110,56,70)(38,108,50,68)(40,106,52,66)(65,88,105,115)(67,86,107,113)(69,84,109,119)(71,82,111,117)>;
G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,84)(34,85)(35,86)(36,87)(37,88)(38,81)(39,82)(40,83)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,113)(56,114)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123), (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), (1,127,65,96)(2,73,66,13)(3,121,67,90)(4,75,68,15)(5,123,69,92)(6,77,70,9)(7,125,71,94)(8,79,72,11)(10,107,78,45)(12,109,80,47)(14,111,74,41)(16,105,76,43)(17,81,28,50)(18,35,29,113)(19,83,30,52)(20,37,31,115)(21,85,32,54)(22,39,25,117)(23,87,26,56)(24,33,27,119)(34,102,120,63)(36,104,114,57)(38,98,116,59)(40,100,118,61)(42,91,112,122)(44,93,106,124)(46,95,108,126)(48,89,110,128)(49,62,88,101)(51,64,82,103)(53,58,84,97)(55,60,86,99), (1,49,43,37)(2,118,44,83)(3,55,45,35)(4,116,46,81)(5,53,47,33)(6,114,48,87)(7,51,41,39)(8,120,42,85)(9,23,128,104)(10,60,121,29)(11,21,122,102)(12,58,123,27)(13,19,124,100)(14,64,125,25)(15,17,126,98)(16,62,127,31)(18,78,99,90)(20,76,101,96)(22,74,103,94)(24,80,97,92)(26,89,57,77)(28,95,59,75)(30,93,61,73)(32,91,63,79)(34,112,54,72)(36,110,56,70)(38,108,50,68)(40,106,52,66)(65,88,105,115)(67,86,107,113)(69,84,109,119)(71,82,111,117) );
G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,103),(26,104),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,84),(34,85),(35,86),(36,87),(37,88),(38,81),(39,82),(40,83),(41,71),(42,72),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,113),(56,114),(73,124),(74,125),(75,126),(76,127),(77,128),(78,121),(79,122),(80,123)], [(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)], [(1,127,65,96),(2,73,66,13),(3,121,67,90),(4,75,68,15),(5,123,69,92),(6,77,70,9),(7,125,71,94),(8,79,72,11),(10,107,78,45),(12,109,80,47),(14,111,74,41),(16,105,76,43),(17,81,28,50),(18,35,29,113),(19,83,30,52),(20,37,31,115),(21,85,32,54),(22,39,25,117),(23,87,26,56),(24,33,27,119),(34,102,120,63),(36,104,114,57),(38,98,116,59),(40,100,118,61),(42,91,112,122),(44,93,106,124),(46,95,108,126),(48,89,110,128),(49,62,88,101),(51,64,82,103),(53,58,84,97),(55,60,86,99)], [(1,49,43,37),(2,118,44,83),(3,55,45,35),(4,116,46,81),(5,53,47,33),(6,114,48,87),(7,51,41,39),(8,120,42,85),(9,23,128,104),(10,60,121,29),(11,21,122,102),(12,58,123,27),(13,19,124,100),(14,64,125,25),(15,17,126,98),(16,62,127,31),(18,78,99,90),(20,76,101,96),(22,74,103,94),(24,80,97,92),(26,89,57,77),(28,95,59,75),(30,93,61,73),(32,91,63,79),(34,112,54,72),(36,110,56,70),(38,108,50,68),(40,106,52,66),(65,88,105,115),(67,86,107,113),(69,84,109,119),(71,82,111,117)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C2.(C8⋊Q8) | C22.7C42 | C22.4Q16 | C23.65C23 | C2×C4.Q8 | C2×C2.D8 | C2×C42.C2 | C4⋊C4 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 8 | 1 | 1 |
Matrix representation of C2.(C8⋊Q8) ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 16 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 16 | 11 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 14 | 11 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[15,0,0,0,0,0,0,8,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,12,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,16,0,0,0,0,16,10,0,0,0,0,0,0,6,16,0,0,0,0,1,11],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,6,14,0,0,0,0,1,11] >;
C2.(C8⋊Q8) in GAP, Magma, Sage, TeX
C_2.(C_8\rtimes Q_8)
% in TeX
G:=Group("C2.(C8:Q8)"); // GroupNames label
G:=SmallGroup(128,791);
// by ID
G=gap.SmallGroup(128,791);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,58,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=a*b^3,d*c*d^-1=a*c^-1>;
// generators/relations