direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×Q8⋊C4, Q8⋊2C42, C42.424D4, (C4×Q8)⋊11C4, C2.2(C4×Q16), C4.109(C4×D4), C4.4(C2×C42), (C2×C4).69Q16, C2.3(C4×SD16), C22.87(C4×D4), C42.261(C2×C4), (C2×C4).129SD16, C23.731(C2×D4), (C22×C4).813D4, C22.21(C2×Q16), C4○3(C22.4Q16), C4.4(C42⋊C2), C22.37(C4○D8), C22.42(C2×SD16), C22.4Q16.53C2, (C22×C8).473C22, (C2×C42).1047C22, (C22×C4).1308C23, C2.4(C23.24D4), (C22×Q8).376C22, (C2×C4×C8).13C2, (C4×C4⋊C4).5C2, (C2×C4×Q8).8C2, C4⋊C4.140(C2×C4), (C2×C8).157(C2×C4), C2.19(C4×C22⋊C4), C2.3(C2×Q8⋊C4), (C2×C4).1303(C2×D4), (C2×Q8).181(C2×C4), (C2×C4).539(C4○D4), (C2×C4⋊C4).747C22, (C2×C4).351(C22×C4), (C2×Q8⋊C4).38C2, (C2×C4).401(C22⋊C4), C22.120(C2×C22⋊C4), SmallGroup(128,493)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×Q8⋊C4
G = < a,b,c,d | a4=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 268 in 166 conjugacy classes, 92 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, Q8⋊C4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C22×Q8, C22.4Q16, C4×C4⋊C4, C2×C4×C8, C2×Q8⋊C4, C2×C4×Q8, C4×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4○D4, Q8⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×SD16, C2×Q16, C4○D8, C4×C22⋊C4, C2×Q8⋊C4, C23.24D4, C4×SD16, C4×Q16, C4×Q8⋊C4
►Regular action on 128 points
Generators in S128
(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 50 45 21)(2 51 46 22)(3 52 47 23)(4 49 48 24)(5 25 125 30)(6 26 126 31)(7 27 127 32)(8 28 128 29)(9 44 16 18)(10 41 13 19)(11 42 14 20)(12 43 15 17)(33 60 38 55)(34 57 39 56)(35 58 40 53)(36 59 37 54)(61 74 82 77)(62 75 83 78)(63 76 84 79)(64 73 81 80)(65 87 70 92)(66 88 71 89)(67 85 72 90)(68 86 69 91)(93 109 114 106)(94 110 115 107)(95 111 116 108)(96 112 113 105)(97 119 102 124)(98 120 103 121)(99 117 104 122)(100 118 101 123)
(1 85 45 90)(2 86 46 91)(3 87 47 92)(4 88 48 89)(5 33 125 38)(6 34 126 39)(7 35 127 40)(8 36 128 37)(9 94 16 115)(10 95 13 116)(11 96 14 113)(12 93 15 114)(17 109 43 106)(18 110 44 107)(19 111 41 108)(20 112 42 105)(21 72 50 67)(22 69 51 68)(23 70 52 65)(24 71 49 66)(25 55 30 60)(26 56 31 57)(27 53 32 58)(28 54 29 59)(61 104 82 99)(62 101 83 100)(63 102 84 97)(64 103 81 98)(73 120 80 121)(74 117 77 122)(75 118 78 123)(76 119 79 124)
(1 120 11 127)(2 117 12 128)(3 118 9 125)(4 119 10 126)(5 47 123 16)(6 48 124 13)(7 45 121 14)(8 46 122 15)(17 29 22 104)(18 30 23 101)(19 31 24 102)(20 32 21 103)(25 52 100 44)(26 49 97 41)(27 50 98 42)(28 51 99 43)(33 65 75 107)(34 66 76 108)(35 67 73 105)(36 68 74 106)(37 69 77 109)(38 70 78 110)(39 71 79 111)(40 72 80 112)(53 85 64 96)(54 86 61 93)(55 87 62 94)(56 88 63 95)(57 89 84 116)(58 90 81 113)(59 91 82 114)(60 92 83 115)
G:=sub<Sym(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), (1,50,45,21)(2,51,46,22)(3,52,47,23)(4,49,48,24)(5,25,125,30)(6,26,126,31)(7,27,127,32)(8,28,128,29)(9,44,16,18)(10,41,13,19)(11,42,14,20)(12,43,15,17)(33,60,38,55)(34,57,39,56)(35,58,40,53)(36,59,37,54)(61,74,82,77)(62,75,83,78)(63,76,84,79)(64,73,81,80)(65,87,70,92)(66,88,71,89)(67,85,72,90)(68,86,69,91)(93,109,114,106)(94,110,115,107)(95,111,116,108)(96,112,113,105)(97,119,102,124)(98,120,103,121)(99,117,104,122)(100,118,101,123), (1,85,45,90)(2,86,46,91)(3,87,47,92)(4,88,48,89)(5,33,125,38)(6,34,126,39)(7,35,127,40)(8,36,128,37)(9,94,16,115)(10,95,13,116)(11,96,14,113)(12,93,15,114)(17,109,43,106)(18,110,44,107)(19,111,41,108)(20,112,42,105)(21,72,50,67)(22,69,51,68)(23,70,52,65)(24,71,49,66)(25,55,30,60)(26,56,31,57)(27,53,32,58)(28,54,29,59)(61,104,82,99)(62,101,83,100)(63,102,84,97)(64,103,81,98)(73,120,80,121)(74,117,77,122)(75,118,78,123)(76,119,79,124), (1,120,11,127)(2,117,12,128)(3,118,9,125)(4,119,10,126)(5,47,123,16)(6,48,124,13)(7,45,121,14)(8,46,122,15)(17,29,22,104)(18,30,23,101)(19,31,24,102)(20,32,21,103)(25,52,100,44)(26,49,97,41)(27,50,98,42)(28,51,99,43)(33,65,75,107)(34,66,76,108)(35,67,73,105)(36,68,74,106)(37,69,77,109)(38,70,78,110)(39,71,79,111)(40,72,80,112)(53,85,64,96)(54,86,61,93)(55,87,62,94)(56,88,63,95)(57,89,84,116)(58,90,81,113)(59,91,82,114)(60,92,83,115)>;
G:=Group( (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), (1,50,45,21)(2,51,46,22)(3,52,47,23)(4,49,48,24)(5,25,125,30)(6,26,126,31)(7,27,127,32)(8,28,128,29)(9,44,16,18)(10,41,13,19)(11,42,14,20)(12,43,15,17)(33,60,38,55)(34,57,39,56)(35,58,40,53)(36,59,37,54)(61,74,82,77)(62,75,83,78)(63,76,84,79)(64,73,81,80)(65,87,70,92)(66,88,71,89)(67,85,72,90)(68,86,69,91)(93,109,114,106)(94,110,115,107)(95,111,116,108)(96,112,113,105)(97,119,102,124)(98,120,103,121)(99,117,104,122)(100,118,101,123), (1,85,45,90)(2,86,46,91)(3,87,47,92)(4,88,48,89)(5,33,125,38)(6,34,126,39)(7,35,127,40)(8,36,128,37)(9,94,16,115)(10,95,13,116)(11,96,14,113)(12,93,15,114)(17,109,43,106)(18,110,44,107)(19,111,41,108)(20,112,42,105)(21,72,50,67)(22,69,51,68)(23,70,52,65)(24,71,49,66)(25,55,30,60)(26,56,31,57)(27,53,32,58)(28,54,29,59)(61,104,82,99)(62,101,83,100)(63,102,84,97)(64,103,81,98)(73,120,80,121)(74,117,77,122)(75,118,78,123)(76,119,79,124), (1,120,11,127)(2,117,12,128)(3,118,9,125)(4,119,10,126)(5,47,123,16)(6,48,124,13)(7,45,121,14)(8,46,122,15)(17,29,22,104)(18,30,23,101)(19,31,24,102)(20,32,21,103)(25,52,100,44)(26,49,97,41)(27,50,98,42)(28,51,99,43)(33,65,75,107)(34,66,76,108)(35,67,73,105)(36,68,74,106)(37,69,77,109)(38,70,78,110)(39,71,79,111)(40,72,80,112)(53,85,64,96)(54,86,61,93)(55,87,62,94)(56,88,63,95)(57,89,84,116)(58,90,81,113)(59,91,82,114)(60,92,83,115) );
G=PermutationGroup([[(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)], [(1,50,45,21),(2,51,46,22),(3,52,47,23),(4,49,48,24),(5,25,125,30),(6,26,126,31),(7,27,127,32),(8,28,128,29),(9,44,16,18),(10,41,13,19),(11,42,14,20),(12,43,15,17),(33,60,38,55),(34,57,39,56),(35,58,40,53),(36,59,37,54),(61,74,82,77),(62,75,83,78),(63,76,84,79),(64,73,81,80),(65,87,70,92),(66,88,71,89),(67,85,72,90),(68,86,69,91),(93,109,114,106),(94,110,115,107),(95,111,116,108),(96,112,113,105),(97,119,102,124),(98,120,103,121),(99,117,104,122),(100,118,101,123)], [(1,85,45,90),(2,86,46,91),(3,87,47,92),(4,88,48,89),(5,33,125,38),(6,34,126,39),(7,35,127,40),(8,36,128,37),(9,94,16,115),(10,95,13,116),(11,96,14,113),(12,93,15,114),(17,109,43,106),(18,110,44,107),(19,111,41,108),(20,112,42,105),(21,72,50,67),(22,69,51,68),(23,70,52,65),(24,71,49,66),(25,55,30,60),(26,56,31,57),(27,53,32,58),(28,54,29,59),(61,104,82,99),(62,101,83,100),(63,102,84,97),(64,103,81,98),(73,120,80,121),(74,117,77,122),(75,118,78,123),(76,119,79,124)], [(1,120,11,127),(2,117,12,128),(3,118,9,125),(4,119,10,126),(5,47,123,16),(6,48,124,13),(7,45,121,14),(8,46,122,15),(17,29,22,104),(18,30,23,101),(19,31,24,102),(20,32,21,103),(25,52,100,44),(26,49,97,41),(27,50,98,42),(28,51,99,43),(33,65,75,107),(34,66,76,108),(35,67,73,105),(36,68,74,106),(37,69,77,109),(38,70,78,110),(39,71,79,111),(40,72,80,112),(53,85,64,96),(54,86,61,93),(55,87,62,94),(56,88,63,95),(57,89,84,116),(58,90,81,113),(59,91,82,114),(60,92,83,115)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4○D4 | C4○D8 |
| kernel | C4×Q8⋊C4 | C22.4Q16 | C4×C4⋊C4 | C2×C4×C8 | C2×Q8⋊C4 | C2×C4×Q8 | Q8⋊C4 | C4×Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C4×Q8⋊C4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 6 | 0 | 0 |
| 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 12 | 5 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 10 | 16 | 0 | 0 |
| 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,4,6,0,0,0,6,13,0,0,0,0,0,12,12,0,0,0,12,5],[4,0,0,0,0,0,10,16,0,0,0,16,7,0,0,0,0,0,13,0,0,0,0,0,4] >;
C4×Q8⋊C4 in GAP, Magma, Sage, TeX
C_4\times Q_8\rtimes C_4
% in TeX
G:=Group("C4xQ8:C4"); // GroupNames label
G:=SmallGroup(128,493);
// by ID
G=gap.SmallGroup(128,493);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,436,2019,248,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations