direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×C4.Q8, C24.191D4, C23.48SD16, C8⋊9(C22×C4), (C22×C8)⋊18C4, C4.1(C22×Q8), (C23×C8).21C2, C4.45(C23×C4), C4⋊C4.346C23, C23.85(C4⋊C4), (C2×C4).183C24, (C2×C8).587C23, C23.838(C2×D4), (C22×C4).603D4, (C22×C4).101Q8, C2.3(C22×SD16), C22.80(C2×SD16), (C22×C8).564C22, (C23×C4).694C22, C22.130(C22×D4), (C22×C4).1504C23, (C2×C8)⋊38(C2×C4), C4.63(C2×C4⋊C4), (C2×C4).841(C2×D4), C2.22(C22×C4⋊C4), C22.74(C2×C4⋊C4), (C2×C4).237(C2×Q8), (C2×C4).149(C4⋊C4), (C22×C4⋊C4).42C2, (C2×C4⋊C4).901C22, (C22×C4).494(C2×C4), (C2×C4).571(C22×C4), SmallGroup(128,1639)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C4.Q8
G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=c-1d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 460 in 300 conjugacy classes, 220 normal (10 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C2×C4.Q8, C22×C4⋊C4, C23×C8, C22×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C24, C4.Q8, C2×C4⋊C4, C2×SD16, C23×C4, C22×D4, C22×Q8, C2×C4.Q8, C22×C4⋊C4, C22×SD16, C22×C4.Q8
►Regular action on 128 points
Generators in S128
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 81)(40 82)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(65 128)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(97 120)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 104)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(17 88 21 84)(18 81 22 85)(19 82 23 86)(20 83 24 87)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)(49 98 53 102)(50 99 54 103)(51 100 55 104)(52 101 56 97)(65 74 69 78)(66 75 70 79)(67 76 71 80)(68 77 72 73)(89 125 93 121)(90 126 94 122)(91 127 95 123)(92 128 96 124)(105 113 109 117)(106 114 110 118)(107 115 111 119)(108 116 112 120)
(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)
(1 117 63 111)(2 120 64 106)(3 115 57 109)(4 118 58 112)(5 113 59 107)(6 116 60 110)(7 119 61 105)(8 114 62 108)(9 104 26 49)(10 99 27 52)(11 102 28 55)(12 97 29 50)(13 100 30 53)(14 103 31 56)(15 98 32 51)(16 101 25 54)(17 74 86 67)(18 77 87 70)(19 80 88 65)(20 75 81 68)(21 78 82 71)(22 73 83 66)(23 76 84 69)(24 79 85 72)(33 121 43 95)(34 124 44 90)(35 127 45 93)(36 122 46 96)(37 125 47 91)(38 128 48 94)(39 123 41 89)(40 126 42 92)
G:=sub<Sym(128)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(65,128)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,104)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,88,21,84)(18,81,22,85)(19,82,23,86)(20,83,24,87)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(65,74,69,78)(66,75,70,79)(67,76,71,80)(68,77,72,73)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124)(105,113,109,117)(106,114,110,118)(107,115,111,119)(108,116,112,120), (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), (1,117,63,111)(2,120,64,106)(3,115,57,109)(4,118,58,112)(5,113,59,107)(6,116,60,110)(7,119,61,105)(8,114,62,108)(9,104,26,49)(10,99,27,52)(11,102,28,55)(12,97,29,50)(13,100,30,53)(14,103,31,56)(15,98,32,51)(16,101,25,54)(17,74,86,67)(18,77,87,70)(19,80,88,65)(20,75,81,68)(21,78,82,71)(22,73,83,66)(23,76,84,69)(24,79,85,72)(33,121,43,95)(34,124,44,90)(35,127,45,93)(36,122,46,96)(37,125,47,91)(38,128,48,94)(39,123,41,89)(40,126,42,92)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(65,128)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,104)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,88,21,84)(18,81,22,85)(19,82,23,86)(20,83,24,87)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(65,74,69,78)(66,75,70,79)(67,76,71,80)(68,77,72,73)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124)(105,113,109,117)(106,114,110,118)(107,115,111,119)(108,116,112,120), (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), (1,117,63,111)(2,120,64,106)(3,115,57,109)(4,118,58,112)(5,113,59,107)(6,116,60,110)(7,119,61,105)(8,114,62,108)(9,104,26,49)(10,99,27,52)(11,102,28,55)(12,97,29,50)(13,100,30,53)(14,103,31,56)(15,98,32,51)(16,101,25,54)(17,74,86,67)(18,77,87,70)(19,80,88,65)(20,75,81,68)(21,78,82,71)(22,73,83,66)(23,76,84,69)(24,79,85,72)(33,121,43,95)(34,124,44,90)(35,127,45,93)(36,122,46,96)(37,125,47,91)(38,128,48,94)(39,123,41,89)(40,126,42,92) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,81),(40,82),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(65,128),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,95),(74,96),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(97,120),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,121),(56,122),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84),(65,110),(66,111),(67,112),(68,105),(69,106),(70,107),(71,108),(72,109),(73,117),(74,118),(75,119),(76,120),(77,113),(78,114),(79,115),(80,116),(89,104),(90,97),(91,98),(92,99),(93,100),(94,101),(95,102),(96,103)], [(1,57,5,61),(2,58,6,62),(3,59,7,63),(4,60,8,64),(9,28,13,32),(10,29,14,25),(11,30,15,26),(12,31,16,27),(17,88,21,84),(18,81,22,85),(19,82,23,86),(20,83,24,87),(33,45,37,41),(34,46,38,42),(35,47,39,43),(36,48,40,44),(49,98,53,102),(50,99,54,103),(51,100,55,104),(52,101,56,97),(65,74,69,78),(66,75,70,79),(67,76,71,80),(68,77,72,73),(89,125,93,121),(90,126,94,122),(91,127,95,123),(92,128,96,124),(105,113,109,117),(106,114,110,118),(107,115,111,119),(108,116,112,120)], [(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)], [(1,117,63,111),(2,120,64,106),(3,115,57,109),(4,118,58,112),(5,113,59,107),(6,116,60,110),(7,119,61,105),(8,114,62,108),(9,104,26,49),(10,99,27,52),(11,102,28,55),(12,97,29,50),(13,100,30,53),(14,103,31,56),(15,98,32,51),(16,101,25,54),(17,74,86,67),(18,77,87,70),(19,80,88,65),(20,75,81,68),(21,78,82,71),(22,73,83,66),(23,76,84,69),(24,79,85,72),(33,121,43,95),(34,124,44,90),(35,127,45,93),(36,122,46,96),(37,125,47,91),(38,128,48,94),(39,123,41,89),(40,126,42,92)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | SD16 |
| kernel | C22×C4.Q8 | C2×C4.Q8 | C22×C4⋊C4 | C23×C8 | C22×C8 | C22×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 3 | 4 | 1 | 16 |
Matrix representation of C22×C4.Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,6,4,0,0,0,0,4,11,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;
C22×C4.Q8 in GAP, Magma, Sage, TeX
C_2^2\times C_4.Q_8
% in TeX
G:=Group("C2^2xC4.Q8"); // GroupNames label
G:=SmallGroup(128,1639);
// by ID
G=gap.SmallGroup(128,1639);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,2804,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^4=c^2,e^2=c^-1*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations