direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C2.C42, C2.1C43, C24.622C23, (C2×C4)⋊4C42, (C2×C42)⋊12C4, C22.71(C4×D4), C22.18(C4×Q8), C23.712(C2×D4), (C22×C4).808D4, (C22×C42).1C2, C23.124(C2×Q8), (C22×C4).106Q8, C2.1(C42⋊4C4), C22.19(C2×C42), C23.335(C4○D4), C23.236(C22×C4), (C23×C4).716C22, C22.39(C42⋊C2), C2.1(C4×C4⋊C4), C2.1(C4×C22⋊C4), C22.40(C2×C4⋊C4), (C2×C4).160(C4⋊C4), (C22×C4).429(C2×C4), C22.82(C2×C22⋊C4), C2.1(C2×C2.C42), (C2×C4).387(C22⋊C4), (C2×C2.C42).32C2, SmallGroup(128,164)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C2.C42
G = < a,b,c,d | a4=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 532 in 376 conjugacy classes, 220 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C23×C4, C23×C4, C2×C2.C42, C22×C42, C4×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C43, C2×C2.C42, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C4×C2.C42
►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 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 46)(18 47)(19 48)(20 45)(21 25)(22 26)(23 27)(24 28)(29 99)(30 100)(31 97)(32 98)(33 77)(34 78)(35 79)(36 80)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 109)(66 110)(67 111)(68 112)(69 73)(70 74)(71 75)(72 76)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(101 105)(102 106)(103 107)(104 108)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(1 85 25 69)(2 86 26 70)(3 87 27 71)(4 88 28 72)(5 81 21 73)(6 82 22 74)(7 83 23 75)(8 84 24 76)(9 95 17 111)(10 96 18 112)(11 93 19 109)(12 94 20 110)(13 91 46 67)(14 92 47 68)(15 89 48 65)(16 90 45 66)(29 37 127 53)(30 38 128 54)(31 39 125 55)(32 40 126 56)(33 113 57 105)(34 114 58 106)(35 115 59 107)(36 116 60 108)(41 123 49 99)(42 124 50 100)(43 121 51 97)(44 122 52 98)(61 101 77 117)(62 102 78 118)(63 103 79 119)(64 104 80 120)
(1 58 45 37)(2 59 46 38)(3 60 47 39)(4 57 48 40)(5 62 20 41)(6 63 17 42)(7 64 18 43)(8 61 19 44)(9 50 22 79)(10 51 23 80)(11 52 24 77)(12 49 21 78)(13 54 26 35)(14 55 27 36)(15 56 28 33)(16 53 25 34)(29 73 114 94)(30 74 115 95)(31 75 116 96)(32 76 113 93)(65 122 88 101)(66 123 85 102)(67 124 86 103)(68 121 87 104)(69 118 90 99)(70 119 91 100)(71 120 92 97)(72 117 89 98)(81 106 110 127)(82 107 111 128)(83 108 112 125)(84 105 109 126)
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), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,46)(18,47)(19,48)(20,45)(21,25)(22,26)(23,27)(24,28)(29,99)(30,100)(31,97)(32,98)(33,77)(34,78)(35,79)(36,80)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,109)(66,110)(67,111)(68,112)(69,73)(70,74)(71,75)(72,76)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (1,85,25,69)(2,86,26,70)(3,87,27,71)(4,88,28,72)(5,81,21,73)(6,82,22,74)(7,83,23,75)(8,84,24,76)(9,95,17,111)(10,96,18,112)(11,93,19,109)(12,94,20,110)(13,91,46,67)(14,92,47,68)(15,89,48,65)(16,90,45,66)(29,37,127,53)(30,38,128,54)(31,39,125,55)(32,40,126,56)(33,113,57,105)(34,114,58,106)(35,115,59,107)(36,116,60,108)(41,123,49,99)(42,124,50,100)(43,121,51,97)(44,122,52,98)(61,101,77,117)(62,102,78,118)(63,103,79,119)(64,104,80,120), (1,58,45,37)(2,59,46,38)(3,60,47,39)(4,57,48,40)(5,62,20,41)(6,63,17,42)(7,64,18,43)(8,61,19,44)(9,50,22,79)(10,51,23,80)(11,52,24,77)(12,49,21,78)(13,54,26,35)(14,55,27,36)(15,56,28,33)(16,53,25,34)(29,73,114,94)(30,74,115,95)(31,75,116,96)(32,76,113,93)(65,122,88,101)(66,123,85,102)(67,124,86,103)(68,121,87,104)(69,118,90,99)(70,119,91,100)(71,120,92,97)(72,117,89,98)(81,106,110,127)(82,107,111,128)(83,108,112,125)(84,105,109,126)>;
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), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,46)(18,47)(19,48)(20,45)(21,25)(22,26)(23,27)(24,28)(29,99)(30,100)(31,97)(32,98)(33,77)(34,78)(35,79)(36,80)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,109)(66,110)(67,111)(68,112)(69,73)(70,74)(71,75)(72,76)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (1,85,25,69)(2,86,26,70)(3,87,27,71)(4,88,28,72)(5,81,21,73)(6,82,22,74)(7,83,23,75)(8,84,24,76)(9,95,17,111)(10,96,18,112)(11,93,19,109)(12,94,20,110)(13,91,46,67)(14,92,47,68)(15,89,48,65)(16,90,45,66)(29,37,127,53)(30,38,128,54)(31,39,125,55)(32,40,126,56)(33,113,57,105)(34,114,58,106)(35,115,59,107)(36,116,60,108)(41,123,49,99)(42,124,50,100)(43,121,51,97)(44,122,52,98)(61,101,77,117)(62,102,78,118)(63,103,79,119)(64,104,80,120), (1,58,45,37)(2,59,46,38)(3,60,47,39)(4,57,48,40)(5,62,20,41)(6,63,17,42)(7,64,18,43)(8,61,19,44)(9,50,22,79)(10,51,23,80)(11,52,24,77)(12,49,21,78)(13,54,26,35)(14,55,27,36)(15,56,28,33)(16,53,25,34)(29,73,114,94)(30,74,115,95)(31,75,116,96)(32,76,113,93)(65,122,88,101)(66,123,85,102)(67,124,86,103)(68,121,87,104)(69,118,90,99)(70,119,91,100)(71,120,92,97)(72,117,89,98)(81,106,110,127)(82,107,111,128)(83,108,112,125)(84,105,109,126) );
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)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,46),(18,47),(19,48),(20,45),(21,25),(22,26),(23,27),(24,28),(29,99),(30,100),(31,97),(32,98),(33,77),(34,78),(35,79),(36,80),(37,41),(38,42),(39,43),(40,44),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,109),(66,110),(67,111),(68,112),(69,73),(70,74),(71,75),(72,76),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(101,105),(102,106),(103,107),(104,108),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128)], [(1,85,25,69),(2,86,26,70),(3,87,27,71),(4,88,28,72),(5,81,21,73),(6,82,22,74),(7,83,23,75),(8,84,24,76),(9,95,17,111),(10,96,18,112),(11,93,19,109),(12,94,20,110),(13,91,46,67),(14,92,47,68),(15,89,48,65),(16,90,45,66),(29,37,127,53),(30,38,128,54),(31,39,125,55),(32,40,126,56),(33,113,57,105),(34,114,58,106),(35,115,59,107),(36,116,60,108),(41,123,49,99),(42,124,50,100),(43,121,51,97),(44,122,52,98),(61,101,77,117),(62,102,78,118),(63,103,79,119),(64,104,80,120)], [(1,58,45,37),(2,59,46,38),(3,60,47,39),(4,57,48,40),(5,62,20,41),(6,63,17,42),(7,64,18,43),(8,61,19,44),(9,50,22,79),(10,51,23,80),(11,52,24,77),(12,49,21,78),(13,54,26,35),(14,55,27,36),(15,56,28,33),(16,53,25,34),(29,73,114,94),(30,74,115,95),(31,75,116,96),(32,76,113,93),(65,122,88,101),(66,123,85,102),(67,124,86,103),(68,121,87,104),(69,118,90,99),(70,119,91,100),(71,120,92,97),(72,117,89,98),(81,106,110,127),(82,107,111,128),(83,108,112,125),(84,105,109,126)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 |
| kernel | C4×C2.C42 | C2×C2.C42 | C22×C42 | C2.C42 | C2×C42 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 4 | 3 | 32 | 24 | 6 | 2 | 8 |
Matrix representation of C4×C2.C42 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 2 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,1,2],[3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,3,0,0,0,0,3] >;
C4×C2.C42 in GAP, Magma, Sage, TeX
C_4\times C_2.C_4^2
% in TeX
G:=Group("C4xC2.C4^2"); // GroupNames label
G:=SmallGroup(128,164);
// by ID
G=gap.SmallGroup(128,164);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,2,112,141,232,352]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations