direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C4⋊C8, C42.672C23, C23.39M4(2), C4⋊2(C22×C8), (C22×C4)⋊9C8, C2.2(C23×C8), (C23×C4).39C4, (C23×C8).11C2, C23.46(C2×C8), (C2×C42).53C4, C4.57(C22×Q8), C23.84(C4⋊C4), (C2×C8).470C23, C42.332(C2×C4), (C2×C4).631C24, C24.136(C2×C4), (C22×C4).821D4, C4.183(C22×D4), (C22×C4).109Q8, (C22×C42).30C2, C22.31(C22×C8), 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
Generators and relations for C22×C4⋊C8
G = < a,b,c,d | a2=b2=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 380 in 320 conjugacy classes, 260 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C24, C4⋊C8, C2×C42, C22×C8, C22×C8, C23×C4, C2×C4⋊C8, C22×C42, C23×C8, C22×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C24, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C23×C4, C22×D4, C22×Q8, C2×C4⋊C8, C22×C4⋊C4, C23×C8, C22×M4(2), C22×C4⋊C8
►Regular action on 128 points
Generators in S128
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 125)(10 126)(11 127)(12 128)(13 121)(14 122)(15 123)(16 124)(17 109)(18 110)(19 111)(20 112)(21 105)(22 106)(23 107)(24 108)(25 120)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 65)(24 66)(25 92)(26 93)(27 94)(28 95)(29 96)(30 89)(31 90)(32 91)(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 75)(2 76 112 116)(3 117 105 77)(4 78 106 118)(5 119 107 79)(6 80 108 120)(7 113 109 73)(8 74 110 114)(9 53 41 37)(10 38 42 54)(11 55 43 39)(12 40 44 56)(13 49 45 33)(14 34 46 50)(15 51 47 35)(16 36 48 52)(17 81 57 26)(18 27 58 82)(19 83 59 28)(20 29 60 84)(21 85 61 30)(22 31 62 86)(23 87 63 32)(24 25 64 88)(65 123 103 91)(66 92 104 124)(67 125 97 93)(68 94 98 126)(69 127 99 95)(70 96 100 128)(71 121 101 89)(72 90 102 122)
(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)(9,125)(10,126)(11,127)(12,128)(13,121)(14,122)(15,123)(16,124)(17,109)(18,110)(19,111)(20,112)(21,105)(22,106)(23,107)(24,108)(25,120)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,93)(42,94)(43,95)(44,96)(45,89)(46,90)(47,91)(48,92)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,92)(26,93)(27,94)(28,95)(29,96)(30,89)(31,90)(32,91)(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), (1,115,111,75)(2,76,112,116)(3,117,105,77)(4,78,106,118)(5,119,107,79)(6,80,108,120)(7,113,109,73)(8,74,110,114)(9,53,41,37)(10,38,42,54)(11,55,43,39)(12,40,44,56)(13,49,45,33)(14,34,46,50)(15,51,47,35)(16,36,48,52)(17,81,57,26)(18,27,58,82)(19,83,59,28)(20,29,60,84)(21,85,61,30)(22,31,62,86)(23,87,63,32)(24,25,64,88)(65,123,103,91)(66,92,104,124)(67,125,97,93)(68,94,98,126)(69,127,99,95)(70,96,100,128)(71,121,101,89)(72,90,102,122), (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)(9,125)(10,126)(11,127)(12,128)(13,121)(14,122)(15,123)(16,124)(17,109)(18,110)(19,111)(20,112)(21,105)(22,106)(23,107)(24,108)(25,120)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,93)(42,94)(43,95)(44,96)(45,89)(46,90)(47,91)(48,92)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,92)(26,93)(27,94)(28,95)(29,96)(30,89)(31,90)(32,91)(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), (1,115,111,75)(2,76,112,116)(3,117,105,77)(4,78,106,118)(5,119,107,79)(6,80,108,120)(7,113,109,73)(8,74,110,114)(9,53,41,37)(10,38,42,54)(11,55,43,39)(12,40,44,56)(13,49,45,33)(14,34,46,50)(15,51,47,35)(16,36,48,52)(17,81,57,26)(18,27,58,82)(19,83,59,28)(20,29,60,84)(21,85,61,30)(22,31,62,86)(23,87,63,32)(24,25,64,88)(65,123,103,91)(66,92,104,124)(67,125,97,93)(68,94,98,126)(69,127,99,95)(70,96,100,128)(71,121,101,89)(72,90,102,122), (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),(9,125),(10,126),(11,127),(12,128),(13,121),(14,122),(15,123),(16,124),(17,109),(18,110),(19,111),(20,112),(21,105),(22,106),(23,107),(24,108),(25,120),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,93),(42,94),(43,95),(44,96),(45,89),(46,90),(47,91),(48,92),(49,101),(50,102),(51,103),(52,104),(53,97),(54,98),(55,99),(56,100),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,65),(24,66),(25,92),(26,93),(27,94),(28,95),(29,96),(30,89),(31,90),(32,91),(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)], [(1,115,111,75),(2,76,112,116),(3,117,105,77),(4,78,106,118),(5,119,107,79),(6,80,108,120),(7,113,109,73),(8,74,110,114),(9,53,41,37),(10,38,42,54),(11,55,43,39),(12,40,44,56),(13,49,45,33),(14,34,46,50),(15,51,47,35),(16,36,48,52),(17,81,57,26),(18,27,58,82),(19,83,59,28),(20,29,60,84),(21,85,61,30),(22,31,62,86),(23,87,63,32),(24,25,64,88),(65,123,103,91),(66,92,104,124),(67,125,97,93),(68,94,98,126),(69,127,99,95),(70,96,100,128),(71,121,101,89),(72,90,102,122)], [(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)]])
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 |
Matrix representation of C22×C4⋊C8 ►in 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] >;
C22×C4⋊C8 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