direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2xC42:4C4, C24.638C23, C23.149C24, C42:40(C2xC4), (C2xC42):14C4, C4o(C42:4C4), (C2xC4).92C42, C4.38(C2xC42), (C22xC42).8C2, C2.2(C22xC42), C22.32(C2xC42), C22.21(C23xC4), C23.350(C4oD4), (C23xC4).637C22, C23.276(C22xC4), (C2xC42).999C22, (C22xC4).1230C23, C22.60(C42:C2), C2.C42.562C22, C4o(C2xC2.C42), (C2xC4)o(C42:4C4), C2.1(C2xC42:C2), C22.47(C2xC4oD4), (C2xC4).480(C22xC4), (C22xC4).449(C2xC4), (C2xC4)o3(C2.C42), (C22xC4)o(C42:4C4), (C2xC2.C42).33C2, (C2xC4)o(C2xC2.C42), SmallGroup(128,999)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2xC42:4C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, cd=dc >
Subgroups: 540 in 420 conjugacy classes, 300 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C2xC4, C2xC4, C23, C23, C42, C22xC4, C22xC4, C24, C2.C42, C2xC42, C23xC4, C23xC4, C2xC2.C42, C42:4C4, C22xC42, C2xC42:4C4
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, C4oD4, C24, C2xC42, C42:C2, C23xC4, C2xC4oD4, C42:4C4, C22xC42, C2xC42:C2, C2xC42:4C4
►Regular action on 128 points
Generators in S128
(1 40)(2 37)(3 38)(4 39)(5 127)(6 128)(7 125)(8 126)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(33 66)(34 67)(35 68)(36 65)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(69 76)(70 73)(71 74)(72 75)(77 89)(78 90)(79 91)(80 92)(81 86)(82 87)(83 88)(84 85)(93 108)(94 105)(95 106)(96 107)(97 104)(98 101)(99 102)(100 103)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)
(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 7 36 102)(2 8 33 103)(3 5 34 104)(4 6 35 101)(9 63 95 78)(10 64 96 79)(11 61 93 80)(12 62 94 77)(13 59 107 91)(14 60 108 92)(15 57 105 89)(16 58 106 90)(17 87 111 55)(18 88 112 56)(19 85 109 53)(20 86 110 54)(21 83 115 51)(22 84 116 52)(23 81 113 49)(24 82 114 50)(25 75 119 46)(26 76 120 47)(27 73 117 48)(28 74 118 45)(29 69 123 42)(30 70 124 43)(31 71 121 44)(32 72 122 41)(37 126 66 100)(38 127 67 97)(39 128 68 98)(40 125 65 99)
(1 87 25 14)(2 56 26 105)(3 85 27 16)(4 54 28 107)(5 109 73 58)(6 20 74 91)(7 111 75 60)(8 18 76 89)(9 38 84 30)(10 68 81 121)(11 40 82 32)(12 66 83 123)(13 35 86 118)(15 33 88 120)(17 46 92 102)(19 48 90 104)(21 69 77 126)(22 43 78 97)(23 71 79 128)(24 41 80 99)(29 94 37 51)(31 96 39 49)(34 53 117 106)(36 55 119 108)(42 62 100 115)(44 64 98 113)(45 59 101 110)(47 57 103 112)(50 122 93 65)(52 124 95 67)(61 125 114 72)(63 127 116 70)
G:=sub<Sym(128)| (1,40)(2,37)(3,38)(4,39)(5,127)(6,128)(7,125)(8,126)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23)(25,32)(26,29)(27,30)(28,31)(33,66)(34,67)(35,68)(36,65)(41,46)(42,47)(43,48)(44,45)(49,54)(50,55)(51,56)(52,53)(57,62)(58,63)(59,64)(60,61)(69,76)(70,73)(71,74)(72,75)(77,89)(78,90)(79,91)(80,92)(81,86)(82,87)(83,88)(84,85)(93,108)(94,105)(95,106)(96,107)(97,104)(98,101)(99,102)(100,103)(109,116)(110,113)(111,114)(112,115)(117,124)(118,121)(119,122)(120,123), (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), (1,7,36,102)(2,8,33,103)(3,5,34,104)(4,6,35,101)(9,63,95,78)(10,64,96,79)(11,61,93,80)(12,62,94,77)(13,59,107,91)(14,60,108,92)(15,57,105,89)(16,58,106,90)(17,87,111,55)(18,88,112,56)(19,85,109,53)(20,86,110,54)(21,83,115,51)(22,84,116,52)(23,81,113,49)(24,82,114,50)(25,75,119,46)(26,76,120,47)(27,73,117,48)(28,74,118,45)(29,69,123,42)(30,70,124,43)(31,71,121,44)(32,72,122,41)(37,126,66,100)(38,127,67,97)(39,128,68,98)(40,125,65,99), (1,87,25,14)(2,56,26,105)(3,85,27,16)(4,54,28,107)(5,109,73,58)(6,20,74,91)(7,111,75,60)(8,18,76,89)(9,38,84,30)(10,68,81,121)(11,40,82,32)(12,66,83,123)(13,35,86,118)(15,33,88,120)(17,46,92,102)(19,48,90,104)(21,69,77,126)(22,43,78,97)(23,71,79,128)(24,41,80,99)(29,94,37,51)(31,96,39,49)(34,53,117,106)(36,55,119,108)(42,62,100,115)(44,64,98,113)(45,59,101,110)(47,57,103,112)(50,122,93,65)(52,124,95,67)(61,125,114,72)(63,127,116,70)>;
G:=Group( (1,40)(2,37)(3,38)(4,39)(5,127)(6,128)(7,125)(8,126)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23)(25,32)(26,29)(27,30)(28,31)(33,66)(34,67)(35,68)(36,65)(41,46)(42,47)(43,48)(44,45)(49,54)(50,55)(51,56)(52,53)(57,62)(58,63)(59,64)(60,61)(69,76)(70,73)(71,74)(72,75)(77,89)(78,90)(79,91)(80,92)(81,86)(82,87)(83,88)(84,85)(93,108)(94,105)(95,106)(96,107)(97,104)(98,101)(99,102)(100,103)(109,116)(110,113)(111,114)(112,115)(117,124)(118,121)(119,122)(120,123), (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), (1,7,36,102)(2,8,33,103)(3,5,34,104)(4,6,35,101)(9,63,95,78)(10,64,96,79)(11,61,93,80)(12,62,94,77)(13,59,107,91)(14,60,108,92)(15,57,105,89)(16,58,106,90)(17,87,111,55)(18,88,112,56)(19,85,109,53)(20,86,110,54)(21,83,115,51)(22,84,116,52)(23,81,113,49)(24,82,114,50)(25,75,119,46)(26,76,120,47)(27,73,117,48)(28,74,118,45)(29,69,123,42)(30,70,124,43)(31,71,121,44)(32,72,122,41)(37,126,66,100)(38,127,67,97)(39,128,68,98)(40,125,65,99), (1,87,25,14)(2,56,26,105)(3,85,27,16)(4,54,28,107)(5,109,73,58)(6,20,74,91)(7,111,75,60)(8,18,76,89)(9,38,84,30)(10,68,81,121)(11,40,82,32)(12,66,83,123)(13,35,86,118)(15,33,88,120)(17,46,92,102)(19,48,90,104)(21,69,77,126)(22,43,78,97)(23,71,79,128)(24,41,80,99)(29,94,37,51)(31,96,39,49)(34,53,117,106)(36,55,119,108)(42,62,100,115)(44,64,98,113)(45,59,101,110)(47,57,103,112)(50,122,93,65)(52,124,95,67)(61,125,114,72)(63,127,116,70) );
G=PermutationGroup([[(1,40),(2,37),(3,38),(4,39),(5,127),(6,128),(7,125),(8,126),(9,16),(10,13),(11,14),(12,15),(17,24),(18,21),(19,22),(20,23),(25,32),(26,29),(27,30),(28,31),(33,66),(34,67),(35,68),(36,65),(41,46),(42,47),(43,48),(44,45),(49,54),(50,55),(51,56),(52,53),(57,62),(58,63),(59,64),(60,61),(69,76),(70,73),(71,74),(72,75),(77,89),(78,90),(79,91),(80,92),(81,86),(82,87),(83,88),(84,85),(93,108),(94,105),(95,106),(96,107),(97,104),(98,101),(99,102),(100,103),(109,116),(110,113),(111,114),(112,115),(117,124),(118,121),(119,122),(120,123)], [(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)], [(1,7,36,102),(2,8,33,103),(3,5,34,104),(4,6,35,101),(9,63,95,78),(10,64,96,79),(11,61,93,80),(12,62,94,77),(13,59,107,91),(14,60,108,92),(15,57,105,89),(16,58,106,90),(17,87,111,55),(18,88,112,56),(19,85,109,53),(20,86,110,54),(21,83,115,51),(22,84,116,52),(23,81,113,49),(24,82,114,50),(25,75,119,46),(26,76,120,47),(27,73,117,48),(28,74,118,45),(29,69,123,42),(30,70,124,43),(31,71,121,44),(32,72,122,41),(37,126,66,100),(38,127,67,97),(39,128,68,98),(40,125,65,99)], [(1,87,25,14),(2,56,26,105),(3,85,27,16),(4,54,28,107),(5,109,73,58),(6,20,74,91),(7,111,75,60),(8,18,76,89),(9,38,84,30),(10,68,81,121),(11,40,82,32),(12,66,83,123),(13,35,86,118),(15,33,88,120),(17,46,92,102),(19,48,90,104),(21,69,77,126),(22,43,78,97),(23,71,79,128),(24,41,80,99),(29,94,37,51),(31,96,39,49),(34,53,117,106),(36,55,119,108),(42,62,100,115),(44,64,98,113),(45,59,101,110),(47,57,103,112),(50,122,93,65),(52,124,95,67),(61,125,114,72),(63,127,116,70)]])
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 |
| type | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | C4oD4 |
| kernel | C2xC42:4C4 | C2xC2.C42 | C42:4C4 | C22xC42 | C2xC42 | C23 |
| # reps | 1 | 4 | 8 | 3 | 48 | 16 |
Matrix representation of C2xC42:4C4 ►in GL5(F5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2],[2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2xC42:4C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_4C_4
% in TeX
G:=Group("C2xC4^2:4C4"); // GroupNames label
G:=SmallGroup(128,999);
// by ID
G=gap.SmallGroup(128,999);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,100]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,c*d=d*c>;
// generators/relations