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, (C22×C4).603D4, C23.838(C2×D4), (C22×C4).101Q8, C2.3(C22×SD16), C22.80(C2×SD16), (C23×C4).694C22, (C22×C8).564C22, C22.130(C22×D4), (C22×C4).1504C23, (C2×C8)⋊38(C2×C4), C4.63(C2×C4⋊C4), (C2×C4).841(C2×D4), C22.74(C2×C4⋊C4), C2.22(C22×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
Subgroups: 460 in 300 conjugacy classes, 220 normal (10 characteristic)
C1, C2, C2 [×14], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×27], C2×C4 [×32], C23 [×15], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×28], C22×C4 [×14], C22×C4 [×20], C24, C4.Q8 [×16], C2×C4⋊C4 [×12], C2×C4⋊C4 [×6], C22×C8 [×14], C23×C4, C23×C4 [×2], C2×C4.Q8 [×12], C22×C4⋊C4 [×2], C23×C8, C22×C4.Q8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], SD16 [×8], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C4.Q8 [×16], C2×C4⋊C4 [×12], C2×SD16 [×12], C23×C4, C22×D4, C22×Q8, C2×C4.Q8 [×12], C22×C4⋊C4, C22×SD16 [×2], C22×C4.Q8
Generators and relations
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 >
►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 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 87)(34 88)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)(105 126)(106 127)(107 128)(108 121)(109 122)(110 123)(111 124)(112 125)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(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)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 102)(90 103)(91 104)(92 97)(93 98)(94 99)(95 100)(96 101)
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)(17 84 21 88)(18 85 22 81)(19 86 23 82)(20 87 24 83)(33 41 37 45)(34 42 38 46)(35 43 39 47)(36 44 40 48)(49 104 53 100)(50 97 54 101)(51 98 55 102)(52 99 56 103)(65 117 69 113)(66 118 70 114)(67 119 71 115)(68 120 72 116)(73 110 77 106)(74 111 78 107)(75 112 79 108)(76 105 80 109)(89 125 93 121)(90 126 94 122)(91 127 95 123)(92 128 96 124)
(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 67)(2 120 64 70)(3 115 57 65)(4 118 58 68)(5 113 59 71)(6 116 60 66)(7 119 61 69)(8 114 62 72)(9 102 28 49)(10 97 29 52)(11 100 30 55)(12 103 31 50)(13 98 32 53)(14 101 25 56)(15 104 26 51)(16 99 27 54)(17 78 82 105)(18 73 83 108)(19 76 84 111)(20 79 85 106)(21 74 86 109)(22 77 87 112)(23 80 88 107)(24 75 81 110)(33 125 47 91)(34 128 48 94)(35 123 41 89)(36 126 42 92)(37 121 43 95)(38 124 44 90)(39 127 45 93)(40 122 46 96)
G:=sub<Sym(128)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,87)(34,88)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113)(105,126)(106,127)(107,128)(108,121)(109,122)(110,123)(111,124)(112,125), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(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)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101), (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29)(17,84,21,88)(18,85,22,81)(19,86,23,82)(20,87,24,83)(33,41,37,45)(34,42,38,46)(35,43,39,47)(36,44,40,48)(49,104,53,100)(50,97,54,101)(51,98,55,102)(52,99,56,103)(65,117,69,113)(66,118,70,114)(67,119,71,115)(68,120,72,116)(73,110,77,106)(74,111,78,107)(75,112,79,108)(76,105,80,109)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124), (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), (1,117,63,67)(2,120,64,70)(3,115,57,65)(4,118,58,68)(5,113,59,71)(6,116,60,66)(7,119,61,69)(8,114,62,72)(9,102,28,49)(10,97,29,52)(11,100,30,55)(12,103,31,50)(13,98,32,53)(14,101,25,56)(15,104,26,51)(16,99,27,54)(17,78,82,105)(18,73,83,108)(19,76,84,111)(20,79,85,106)(21,74,86,109)(22,77,87,112)(23,80,88,107)(24,75,81,110)(33,125,47,91)(34,128,48,94)(35,123,41,89)(36,126,42,92)(37,121,43,95)(38,124,44,90)(39,127,45,93)(40,122,46,96)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,87)(34,88)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113)(105,126)(106,127)(107,128)(108,121)(109,122)(110,123)(111,124)(112,125), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(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)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101), (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29)(17,84,21,88)(18,85,22,81)(19,86,23,82)(20,87,24,83)(33,41,37,45)(34,42,38,46)(35,43,39,47)(36,44,40,48)(49,104,53,100)(50,97,54,101)(51,98,55,102)(52,99,56,103)(65,117,69,113)(66,118,70,114)(67,119,71,115)(68,120,72,116)(73,110,77,106)(74,111,78,107)(75,112,79,108)(76,105,80,109)(89,125,93,121)(90,126,94,122)(91,127,95,123)(92,128,96,124), (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), (1,117,63,67)(2,120,64,70)(3,115,57,65)(4,118,58,68)(5,113,59,71)(6,116,60,66)(7,119,61,69)(8,114,62,72)(9,102,28,49)(10,97,29,52)(11,100,30,55)(12,103,31,50)(13,98,32,53)(14,101,25,56)(15,104,26,51)(16,99,27,54)(17,78,82,105)(18,73,83,108)(19,76,84,111)(20,79,85,106)(21,74,86,109)(22,77,87,112)(23,80,88,107)(24,75,81,110)(33,125,47,91)(34,128,48,94)(35,123,41,89)(36,126,42,92)(37,121,43,95)(38,124,44,90)(39,127,45,93)(40,122,46,96) );
G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(33,87),(34,88),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68),(73,95),(74,96),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,113),(105,126),(106,127),(107,128),(108,121),(109,122),(110,123),(111,124),(112,125)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(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),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(73,117),(74,118),(75,119),(76,120),(77,113),(78,114),(79,115),(80,116),(89,102),(90,103),(91,104),(92,97),(93,98),(94,99),(95,100),(96,101)], [(1,57,5,61),(2,58,6,62),(3,59,7,63),(4,60,8,64),(9,30,13,26),(10,31,14,27),(11,32,15,28),(12,25,16,29),(17,84,21,88),(18,85,22,81),(19,86,23,82),(20,87,24,83),(33,41,37,45),(34,42,38,46),(35,43,39,47),(36,44,40,48),(49,104,53,100),(50,97,54,101),(51,98,55,102),(52,99,56,103),(65,117,69,113),(66,118,70,114),(67,119,71,115),(68,120,72,116),(73,110,77,106),(74,111,78,107),(75,112,79,108),(76,105,80,109),(89,125,93,121),(90,126,94,122),(91,127,95,123),(92,128,96,124)], [(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)], [(1,117,63,67),(2,120,64,70),(3,115,57,65),(4,118,58,68),(5,113,59,71),(6,116,60,66),(7,119,61,69),(8,114,62,72),(9,102,28,49),(10,97,29,52),(11,100,30,55),(12,103,31,50),(13,98,32,53),(14,101,25,56),(15,104,26,51),(16,99,27,54),(17,78,82,105),(18,73,83,108),(19,76,84,111),(20,79,85,106),(21,74,86,109),(22,77,87,112),(23,80,88,107),(24,75,81,110),(33,125,47,91),(34,128,48,94),(35,123,41,89),(36,126,42,92),(37,121,43,95),(38,124,44,90),(39,127,45,93),(40,122,46,96)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀