p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2.D8⋊4C4, C2.8(C4×D8), C4⋊C4.11Q8, C4.23(C4×Q8), (C2×C4).107D8, C22.41(C2×D8), C2.4(Q8.Q8), C2.3(D4⋊Q8), C23.776(C2×D4), C22.156(C4×D4), (C22×C4).691D4, C2.8(Q16⋊C4), C22.61(C4○D8), C22.4Q16.7C2, (C22×C8).42C22, C4.11(C42.C2), C4.22(C42⋊C2), C4.13(C42⋊2C2), (C2×C42).291C22, C2.4(C22.D8), C22.75(C22⋊Q8), (C22×C4).1374C23, C2.4(C23.20D4), C22.70(C8.C22), C22.7C42.20C2, C23.65C23.4C2, C2.9(C23.63C23), C22.88(C22.D4), (C4×C4⋊C4).14C2, C4⋊C4.76(C2×C4), (C2×C8).38(C2×C4), (C2×C2.D8).5C2, (C2×C4).269(C2×Q8), (C2×C4).571(C4○D4), (C2×C4⋊C4).769C22, (C2×C4).392(C22×C4), SmallGroup(128,650)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.D8⋊4C4
G = < a,b,c,d | a2=b8=d4=1, c2=a, dbd-1=ab=ba, dcd-1=ac=ca, ad=da, cbc-1=b-1 >
Subgroups: 220 in 115 conjugacy classes, 58 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, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.65C23, C2×C2.D8, C2.D8⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D8, C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×D8, C4○D8, C8.C22, C23.63C23, C4×D8, Q16⋊C4, D4⋊Q8, Q8.Q8, C22.D8, C23.20D4, C2.D8⋊4C4
►Regular action on 128 points
Generators in S128
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 65)(56 66)(113 124)(114 125)(115 126)(116 127)(117 128)(118 121)(119 122)(120 123)
(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)
(1 110 64 14)(2 109 57 13)(3 108 58 12)(4 107 59 11)(5 106 60 10)(6 105 61 9)(7 112 62 16)(8 111 63 15)(17 87 97 45)(18 86 98 44)(19 85 99 43)(20 84 100 42)(21 83 101 41)(22 82 102 48)(23 81 103 47)(24 88 104 46)(25 79 89 37)(26 78 90 36)(27 77 91 35)(28 76 92 34)(29 75 93 33)(30 74 94 40)(31 73 95 39)(32 80 96 38)(49 120 67 123)(50 119 68 122)(51 118 69 121)(52 117 70 128)(53 116 71 127)(54 115 72 126)(55 114 65 125)(56 113 66 124)
(1 74 123 99)(2 33 124 20)(3 76 125 101)(4 35 126 22)(5 78 127 103)(6 37 128 24)(7 80 121 97)(8 39 122 18)(9 25 70 88)(10 90 71 47)(11 27 72 82)(12 92 65 41)(13 29 66 84)(14 94 67 43)(15 31 68 86)(16 96 69 45)(17 62 38 118)(19 64 40 120)(21 58 34 114)(23 60 36 116)(26 53 81 106)(28 55 83 108)(30 49 85 110)(32 51 87 112)(42 109 93 56)(44 111 95 50)(46 105 89 52)(48 107 91 54)(57 75 113 100)(59 77 115 102)(61 79 117 104)(63 73 119 98)
G:=sub<Sym(128)| (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,110,64,14)(2,109,57,13)(3,108,58,12)(4,107,59,11)(5,106,60,10)(6,105,61,9)(7,112,62,16)(8,111,63,15)(17,87,97,45)(18,86,98,44)(19,85,99,43)(20,84,100,42)(21,83,101,41)(22,82,102,48)(23,81,103,47)(24,88,104,46)(25,79,89,37)(26,78,90,36)(27,77,91,35)(28,76,92,34)(29,75,93,33)(30,74,94,40)(31,73,95,39)(32,80,96,38)(49,120,67,123)(50,119,68,122)(51,118,69,121)(52,117,70,128)(53,116,71,127)(54,115,72,126)(55,114,65,125)(56,113,66,124), (1,74,123,99)(2,33,124,20)(3,76,125,101)(4,35,126,22)(5,78,127,103)(6,37,128,24)(7,80,121,97)(8,39,122,18)(9,25,70,88)(10,90,71,47)(11,27,72,82)(12,92,65,41)(13,29,66,84)(14,94,67,43)(15,31,68,86)(16,96,69,45)(17,62,38,118)(19,64,40,120)(21,58,34,114)(23,60,36,116)(26,53,81,106)(28,55,83,108)(30,49,85,110)(32,51,87,112)(42,109,93,56)(44,111,95,50)(46,105,89,52)(48,107,91,54)(57,75,113,100)(59,77,115,102)(61,79,117,104)(63,73,119,98)>;
G:=Group( (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,110,64,14)(2,109,57,13)(3,108,58,12)(4,107,59,11)(5,106,60,10)(6,105,61,9)(7,112,62,16)(8,111,63,15)(17,87,97,45)(18,86,98,44)(19,85,99,43)(20,84,100,42)(21,83,101,41)(22,82,102,48)(23,81,103,47)(24,88,104,46)(25,79,89,37)(26,78,90,36)(27,77,91,35)(28,76,92,34)(29,75,93,33)(30,74,94,40)(31,73,95,39)(32,80,96,38)(49,120,67,123)(50,119,68,122)(51,118,69,121)(52,117,70,128)(53,116,71,127)(54,115,72,126)(55,114,65,125)(56,113,66,124), (1,74,123,99)(2,33,124,20)(3,76,125,101)(4,35,126,22)(5,78,127,103)(6,37,128,24)(7,80,121,97)(8,39,122,18)(9,25,70,88)(10,90,71,47)(11,27,72,82)(12,92,65,41)(13,29,66,84)(14,94,67,43)(15,31,68,86)(16,96,69,45)(17,62,38,118)(19,64,40,120)(21,58,34,114)(23,60,36,116)(26,53,81,106)(28,55,83,108)(30,49,85,110)(32,51,87,112)(42,109,93,56)(44,111,95,50)(46,105,89,52)(48,107,91,54)(57,75,113,100)(59,77,115,102)(61,79,117,104)(63,73,119,98) );
G=PermutationGroup([[(1,64),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,65),(56,66),(113,124),(114,125),(115,126),(116,127),(117,128),(118,121),(119,122),(120,123)], [(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)], [(1,110,64,14),(2,109,57,13),(3,108,58,12),(4,107,59,11),(5,106,60,10),(6,105,61,9),(7,112,62,16),(8,111,63,15),(17,87,97,45),(18,86,98,44),(19,85,99,43),(20,84,100,42),(21,83,101,41),(22,82,102,48),(23,81,103,47),(24,88,104,46),(25,79,89,37),(26,78,90,36),(27,77,91,35),(28,76,92,34),(29,75,93,33),(30,74,94,40),(31,73,95,39),(32,80,96,38),(49,120,67,123),(50,119,68,122),(51,118,69,121),(52,117,70,128),(53,116,71,127),(54,115,72,126),(55,114,65,125),(56,113,66,124)], [(1,74,123,99),(2,33,124,20),(3,76,125,101),(4,35,126,22),(5,78,127,103),(6,37,128,24),(7,80,121,97),(8,39,122,18),(9,25,70,88),(10,90,71,47),(11,27,72,82),(12,92,65,41),(13,29,66,84),(14,94,67,43),(15,31,68,86),(16,96,69,45),(17,62,38,118),(19,64,40,120),(21,58,34,114),(23,60,36,116),(26,53,81,106),(28,55,83,108),(30,49,85,110),(32,51,87,112),(42,109,93,56),(44,111,95,50),(46,105,89,52),(48,107,91,54),(57,75,113,100),(59,77,115,102),(61,79,117,104),(63,73,119,98)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D8 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C2.D8⋊4C4 | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.65C23 | C2×C2.D8 | C2.D8 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C2.D8⋊4C4 ►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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 16 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 4 |
| 0 | 0 | 0 | 0 | 13 | 10 |
| 13 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,0,11,1,0,0,0,0,16,6],[0,15,0,0,0,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,2,0,0,0,0,0,0,0,7,13,0,0,0,0,4,10],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2.D8⋊4C4 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes_4C_4
% in TeX
G:=Group("C2.D8:4C4"); // GroupNames label
G:=SmallGroup(128,650);
// by ID
G=gap.SmallGroup(128,650);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=a,d*b*d^-1=a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c^-1=b^-1>;
// generators/relations