direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22.7C42, C23.41C42, C23.36M4(2), (C22×C8)⋊9C4, (C22×C4)⋊5C8, (C23×C8).1C2, (C23×C4).25C4, C23.43(C2×C8), (C2×C42).29C4, C22.16(C4×C8), (C2×C4).86C42, C23.81(C4⋊C4), C22.22(C4⋊C8), C24.133(C2×C4), (C22×C4).811D4, (C22×C42).4C2, (C22×C4).107Q8, C22.17(C22×C8), C22.25(C2×C42), C22.14(C8⋊C4), C22.42(C22⋊C8), C4○(C22.7C42), (C22×C8).464C22, C23.244(C22×C4), (C23×C4).717C22, (C2×C42).982C22, C22.32(C2×M4(2)), C23.224(C22⋊C4), C4.30(C2.C42), (C22×C4).1597C23, C22.25(C2.C42), C2.4(C2×C4×C8), C2.1(C2×C4⋊C8), (C2×C4)⋊10(C2×C8), (C2×C8)⋊34(C2×C4), C4.69(C2×C4⋊C4), C2.3(C2×C8⋊C4), C2.1(C2×C22⋊C8), C22.49(C2×C4⋊C4), C4.99(C2×C22⋊C4), (C2×C4).326(C2×Q8), (C2×C4).162(C4⋊C4), (C2×C4).1488(C2×D4), (C2×C4).587(C22×C4), (C22×C4).431(C2×C4), C2.2(C2×C2.C42), (C2×C4).391(C22⋊C4), C22.100(C2×C22⋊C4), (C2×C4)○(C22.7C42), (C22×C4)○(C22.7C42), SmallGroup(128,459)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.7C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >
Subgroups: 356 in 264 conjugacy classes, 172 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2×C42, C2×C42, C22×C8, C22×C8, C23×C4, C23×C4, C22.7C42, C22×C42, C23×C8, C2×C22.7C42
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×M4(2), C22.7C42, C2×C2.C42, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8, C2×C4⋊C8, C2×C22.7C42
►Regular action on 128 points
Generators in S128
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 121)(16 122)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(96 116)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 65)(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)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 121)(88 122)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(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)
(1 119 111 73)(2 48 112 10)(3 113 105 75)(4 42 106 12)(5 115 107 77)(6 44 108 14)(7 117 109 79)(8 46 110 16)(9 49 47 39)(11 51 41 33)(13 53 43 35)(15 55 45 37)(17 121 97 26)(18 88 98 90)(19 123 99 28)(20 82 100 92)(21 125 101 30)(22 84 102 94)(23 127 103 32)(24 86 104 96)(25 65 128 64)(27 67 122 58)(29 69 124 60)(31 71 126 62)(34 76 52 114)(36 78 54 116)(38 80 56 118)(40 74 50 120)(57 89 66 87)(59 91 68 81)(61 93 70 83)(63 95 72 85)
G:=sub<Sym(128)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,121)(16,122)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,117)(90,118)(91,119)(92,120)(93,113)(94,114)(95,115)(96,116), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(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)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,119,111,73)(2,48,112,10)(3,113,105,75)(4,42,106,12)(5,115,107,77)(6,44,108,14)(7,117,109,79)(8,46,110,16)(9,49,47,39)(11,51,41,33)(13,53,43,35)(15,55,45,37)(17,121,97,26)(18,88,98,90)(19,123,99,28)(20,82,100,92)(21,125,101,30)(22,84,102,94)(23,127,103,32)(24,86,104,96)(25,65,128,64)(27,67,122,58)(29,69,124,60)(31,71,126,62)(34,76,52,114)(36,78,54,116)(38,80,56,118)(40,74,50,120)(57,89,66,87)(59,91,68,81)(61,93,70,83)(63,95,72,85)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,121)(16,122)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,117)(90,118)(91,119)(92,120)(93,113)(94,114)(95,115)(96,116), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(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)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (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), (1,119,111,73)(2,48,112,10)(3,113,105,75)(4,42,106,12)(5,115,107,77)(6,44,108,14)(7,117,109,79)(8,46,110,16)(9,49,47,39)(11,51,41,33)(13,53,43,35)(15,55,45,37)(17,121,97,26)(18,88,98,90)(19,123,99,28)(20,82,100,92)(21,125,101,30)(22,84,102,94)(23,127,103,32)(24,86,104,96)(25,65,128,64)(27,67,122,58)(29,69,124,60)(31,71,126,62)(34,76,52,114)(36,78,54,116)(38,80,56,118)(40,74,50,120)(57,89,66,87)(59,91,68,81)(61,93,70,83)(63,95,72,85) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,121),(16,122),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,97),(56,98),(65,108),(66,109),(67,110),(68,111),(69,112),(70,105),(71,106),(72,107),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(89,117),(90,118),(91,119),(92,120),(93,113),(94,114),(95,115),(96,116)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,65),(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),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,121),(88,122)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128)], [(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)], [(1,119,111,73),(2,48,112,10),(3,113,105,75),(4,42,106,12),(5,115,107,77),(6,44,108,14),(7,117,109,79),(8,46,110,16),(9,49,47,39),(11,51,41,33),(13,53,43,35),(15,55,45,37),(17,121,97,26),(18,88,98,90),(19,123,99,28),(20,82,100,92),(21,125,101,30),(22,84,102,94),(23,127,103,32),(24,86,104,96),(25,65,128,64),(27,67,122,58),(29,69,124,60),(31,71,126,62),(34,76,52,114),(36,78,54,116),(38,80,56,118),(40,74,50,120),(57,89,66,87),(59,91,68,81),(61,93,70,83),(63,95,72,85)]])
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 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C2×C22.7C42 | C22.7C42 | C22×C42 | C23×C8 | C2×C42 | C22×C8 | C23×C4 | C22×C4 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 32 | 6 | 2 | 8 |
Matrix representation of C2×C22.7C42 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 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 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 | 8 |
| 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[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],[2,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,14,3,0,0,0,8,3],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,15,1] >;
C2×C22.7C42 in GAP, Magma, Sage, TeX
C_2\times C_2^2._7C_4^2
% in TeX
G:=Group("C2xC2^2.7C4^2"); // GroupNames label
G:=SmallGroup(128,459);
// by ID
G=gap.SmallGroup(128,459);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c>;
// generators/relations