direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C4⋊C8, C42.672C23, C23.39M4(2), (C22×C4)⋊9C8, C4⋊2(C22×C8), C2.2(C23×C8), C23.46(C2×C8), (C23×C8).11C2, (C23×C4).39C4, (C2×C42).53C4, C4.57(C22×Q8), C23.84(C4⋊C4), C42.332(C2×C4), C24.136(C2×C4), (C2×C8).470C23, (C2×C4).631C24, C4.183(C22×D4), (C22×C4).821D4, (C22×C4).109Q8, C22.31(C22×C8), (C22×C42).30C2, C22.38(C23×C4), C2.4(C22×M4(2)), (C22×C8).504C22, C23.290(C22×C4), (C23×C4).719C22, C22.60(C2×M4(2)), (C2×C42).1102C22, (C22×C4).1649C23, C4○(C2×C4⋊C8), (C2×C4)○2(C4⋊C8), (C2×C4)⋊11(C2×C8), C4.84(C2×C4⋊C4), (C22×C4)○(C4⋊C8), C2.3(C22×C4⋊C4), C22.73(C2×C4⋊C4), (C2×C4).355(C2×Q8), (C2×C4).168(C4⋊C4), (C2×C4).1565(C2×D4), (C22×C4).493(C2×C4), (C2×C4).625(C22×C4), (C2×C4)○(C2×C4⋊C8), (C22×C4)○(C2×C4⋊C8), SmallGroup(128,1634)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 320 conjugacy classes, 260 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×14], C4 [×4], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×59], C2×C4 [×12], C23 [×15], C42 [×16], C2×C8 [×8], C2×C8 [×24], C22×C4 [×34], C22×C4 [×4], C24, C4⋊C8 [×16], C2×C42 [×12], C22×C8 [×12], C22×C8 [×8], C23×C4 [×3], C2×C4⋊C8 [×12], C22×C42, C23×C8 [×2], C22×C4⋊C8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C4⋊C8 [×16], C2×C4⋊C4 [×12], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C22×D4, C22×Q8, C2×C4⋊C8 [×12], C22×C4⋊C4, C23×C8, C22×M4(2), C22×C4⋊C8
Generators and relations
G = < a,b,c,d | a2=b2=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 84)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 113)(96 114)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 96)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)
(1 115 111 79)(2 80 112 116)(3 117 105 73)(4 74 106 118)(5 119 107 75)(6 76 108 120)(7 113 109 77)(8 78 110 114)(9 53 45 33)(10 34 46 54)(11 55 47 35)(12 36 48 56)(13 49 41 37)(14 38 42 50)(15 51 43 39)(16 40 44 52)(17 85 61 91)(18 92 62 86)(19 87 63 93)(20 94 64 88)(21 81 57 95)(22 96 58 82)(23 83 59 89)(24 90 60 84)(25 98 126 66)(26 67 127 99)(27 100 128 68)(28 69 121 101)(29 102 122 70)(30 71 123 103)(31 104 124 72)(32 65 125 97)
(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)
G:=sub<Sym(128)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124), (1,115,111,79)(2,80,112,116)(3,117,105,73)(4,74,106,118)(5,119,107,75)(6,76,108,120)(7,113,109,77)(8,78,110,114)(9,53,45,33)(10,34,46,54)(11,55,47,35)(12,36,48,56)(13,49,41,37)(14,38,42,50)(15,51,43,39)(16,40,44,52)(17,85,61,91)(18,92,62,86)(19,87,63,93)(20,94,64,88)(21,81,57,95)(22,96,58,82)(23,83,59,89)(24,90,60,84)(25,98,126,66)(26,67,127,99)(27,100,128,68)(28,69,121,101)(29,102,122,70)(30,71,123,103)(31,104,124,72)(32,65,125,97), (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)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124), (1,115,111,79)(2,80,112,116)(3,117,105,73)(4,74,106,118)(5,119,107,75)(6,76,108,120)(7,113,109,77)(8,78,110,114)(9,53,45,33)(10,34,46,54)(11,55,47,35)(12,36,48,56)(13,49,41,37)(14,38,42,50)(15,51,43,39)(16,40,44,52)(17,85,61,91)(18,92,62,86)(19,87,63,93)(20,94,64,88)(21,81,57,95)(22,96,58,82)(23,83,59,89)(24,90,60,84)(25,98,126,66)(26,67,127,99)(27,100,128,68)(28,69,121,101)(29,102,122,70)(30,71,123,103)(31,104,124,72)(32,65,125,97), (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) );
G=PermutationGroup([(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41),(33,69),(34,70),(35,71),(36,72),(37,65),(38,66),(39,67),(40,68),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(73,85),(74,86),(75,87),(76,88),(77,81),(78,82),(79,83),(80,84),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,113),(96,114)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(25,96),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(81,125),(82,126),(83,127),(84,128),(85,121),(86,122),(87,123),(88,124)], [(1,115,111,79),(2,80,112,116),(3,117,105,73),(4,74,106,118),(5,119,107,75),(6,76,108,120),(7,113,109,77),(8,78,110,114),(9,53,45,33),(10,34,46,54),(11,55,47,35),(12,36,48,56),(13,49,41,37),(14,38,42,50),(15,51,43,39),(16,40,44,52),(17,85,61,91),(18,92,62,86),(19,87,63,93),(20,94,64,88),(21,81,57,95),(22,96,58,82),(23,83,59,89),(24,90,60,84),(25,98,126,66),(26,67,127,99),(27,100,128,68),(28,69,121,101),(29,102,122,70),(30,71,123,103),(31,104,124,72),(32,65,125,97)], [(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)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,1,0,0,0,15,1],[2,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,16] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4AF | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C22×C4⋊C8 | C2×C4⋊C8 | C22×C42 | C23×C8 | C2×C42 | C23×C4 | C22×C4 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 12 | 1 | 2 | 12 | 4 | 32 | 4 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4\rtimes C_8
% in TeX
G:=Group("C2^2xC4:C8"); // GroupNames label
G:=SmallGroup(128,1634);
// by ID
G=gap.SmallGroup(128,1634);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀