direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C2.Q32, C23.58D8, C22.5Q32, C22.11SD32, (C2×Q16)⋊9C4, Q16⋊7(C2×C4), (C2×C4).78D8, C8.91(C2×D4), C2.1(C2×Q32), (C2×C8).247D4, C2.2(C2×SD32), C4.9(C2×SD16), (C22×C16).7C2, C8.28(C22×C4), (C2×C4).76SD16, C22.52(C2×D8), C8.25(C22⋊C4), (C2×C8).493C23, (C2×C16).62C22, (C22×C4).583D4, (C22×Q16).6C2, C4.24(D4⋊C4), (C2×Q16).98C22, C2.D8.142C22, (C22×C8).527C22, C22.53(D4⋊C4), (C2×C8).175(C2×C4), (C2×C4).755(C2×D4), C4.49(C2×C22⋊C4), (C2×C2.D8).22C2, C2.27(C2×D4⋊C4), (C2×C4).270(C22⋊C4), SmallGroup(128,869)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C2.Q32
G = < a,b,c,d | a2=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 244 in 116 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C16, C4⋊C4, C2×C8, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×Q8, C2.D8, C2.D8, C2×C16, C2×C16, C2×C4⋊C4, C22×C8, C2×Q16, C2×Q16, C22×Q8, C2.Q32, C2×C2.D8, C22×C16, C22×Q16, C2×C2.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, SD32, Q32, C2×C22⋊C4, C2×D8, C2×SD16, C2.Q32, C2×D4⋊C4, C2×SD32, C2×Q32, C2×C2.Q32
►Regular action on 128 points
Generators in S128
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 65)(61 66)(62 67)(63 68)(64 69)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 113)
(1 93)(2 94)(3 95)(4 96)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 124)(50 125)(51 126)(52 127)(53 128)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 97)(77 98)(78 99)(79 100)(80 101)
(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 76 9 68)(2 112 10 104)(3 74 11 66)(4 110 12 102)(5 72 13 80)(6 108 14 100)(7 70 15 78)(8 106 16 98)(17 116 25 124)(18 56 26 64)(19 114 27 122)(20 54 28 62)(21 128 29 120)(22 52 30 60)(23 126 31 118)(24 50 32 58)(33 49 41 57)(34 123 42 115)(35 63 43 55)(36 121 44 113)(37 61 45 53)(38 119 46 127)(39 59 47 51)(40 117 48 125)(65 96 73 88)(67 94 75 86)(69 92 77 84)(71 90 79 82)(81 109 89 101)(83 107 91 99)(85 105 93 97)(87 103 95 111)
G:=sub<Sym(128)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,65)(61,66)(62,67)(63,68)(64,69)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,113), (1,93)(2,94)(3,95)(4,96)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,124)(50,125)(51,126)(52,127)(53,128)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,97)(77,98)(78,99)(79,100)(80,101), (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), (1,76,9,68)(2,112,10,104)(3,74,11,66)(4,110,12,102)(5,72,13,80)(6,108,14,100)(7,70,15,78)(8,106,16,98)(17,116,25,124)(18,56,26,64)(19,114,27,122)(20,54,28,62)(21,128,29,120)(22,52,30,60)(23,126,31,118)(24,50,32,58)(33,49,41,57)(34,123,42,115)(35,63,43,55)(36,121,44,113)(37,61,45,53)(38,119,46,127)(39,59,47,51)(40,117,48,125)(65,96,73,88)(67,94,75,86)(69,92,77,84)(71,90,79,82)(81,109,89,101)(83,107,91,99)(85,105,93,97)(87,103,95,111)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,65)(61,66)(62,67)(63,68)(64,69)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,113), (1,93)(2,94)(3,95)(4,96)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,124)(50,125)(51,126)(52,127)(53,128)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,97)(77,98)(78,99)(79,100)(80,101), (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), (1,76,9,68)(2,112,10,104)(3,74,11,66)(4,110,12,102)(5,72,13,80)(6,108,14,100)(7,70,15,78)(8,106,16,98)(17,116,25,124)(18,56,26,64)(19,114,27,122)(20,54,28,62)(21,128,29,120)(22,52,30,60)(23,126,31,118)(24,50,32,58)(33,49,41,57)(34,123,42,115)(35,63,43,55)(36,121,44,113)(37,61,45,53)(38,119,46,127)(39,59,47,51)(40,117,48,125)(65,96,73,88)(67,94,75,86)(69,92,77,84)(71,90,79,82)(81,109,89,101)(83,107,91,99)(85,105,93,97)(87,103,95,111) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,65),(61,66),(62,67),(63,68),(64,69),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,121),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,113)], [(1,93),(2,94),(3,95),(4,96),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(49,124),(50,125),(51,126),(52,127),(53,128),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122),(64,123),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,97),(77,98),(78,99),(79,100),(80,101)], [(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)], [(1,76,9,68),(2,112,10,104),(3,74,11,66),(4,110,12,102),(5,72,13,80),(6,108,14,100),(7,70,15,78),(8,106,16,98),(17,116,25,124),(18,56,26,64),(19,114,27,122),(20,54,28,62),(21,128,29,120),(22,52,30,60),(23,126,31,118),(24,50,32,58),(33,49,41,57),(34,123,42,115),(35,63,43,55),(36,121,44,113),(37,61,45,53),(38,119,46,127),(39,59,47,51),(40,117,48,125),(65,96,73,88),(67,94,75,86),(69,92,77,84),(71,90,79,82),(81,109,89,101),(83,107,91,99),(85,105,93,97),(87,103,95,111)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D8 | SD32 | Q32 |
| kernel | C2×C2.Q32 | C2.Q32 | C2×C2.D8 | C22×C16 | C22×Q16 | C2×Q16 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 8 |
Matrix representation of C2×C2.Q32 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 6 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,13,6,0,0,11,13],[16,0,0,0,0,16,0,0,0,0,5,12,0,0,12,12] >;
C2×C2.Q32 in GAP, Magma, Sage, TeX
C_2\times C_2.Q_{32} % in TeX
G:=Group("C2xC2.Q32"); // GroupNames label
G:=SmallGroup(128,869);
// by ID
G=gap.SmallGroup(128,869);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations