direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22×Q32, C8.11C24, C23.65D8, C16.12C23, Q16.1C23, C4.23(C2×D8), (C2×C4).96D8, C8.56(C2×D4), (C2×C8).264D4, C4.17(C22×D4), C2.26(C22×D8), C22.77(C2×D8), (C22×C16).12C2, (C2×C8).573C23, (C2×C16).91C22, (C22×C4).623D4, (C22×Q16).10C2, (C22×C8).543C22, (C2×Q16).146C22, (C2×C4).874(C2×D4), SmallGroup(128,2142)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×Q32
G = < a,b,c,d | a2=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 340 in 180 conjugacy classes, 100 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C16, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×Q8, C2×C16, Q32, C22×C8, C2×Q16, C2×Q16, C22×Q8, C22×C16, C2×Q32, C22×Q16, C22×Q32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, Q32, C2×D8, C22×D4, C2×Q32, C22×D8, C22×Q32
►Regular action on 128 points
Generators in S128
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 97)(32 98)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 126)(50 127)(51 128)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)
(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 28 9 20)(2 27 10 19)(3 26 11 18)(4 25 12 17)(5 24 13 32)(6 23 14 31)(7 22 15 30)(8 21 16 29)(33 118 41 126)(34 117 42 125)(35 116 43 124)(36 115 44 123)(37 114 45 122)(38 113 46 121)(39 128 47 120)(40 127 48 119)(49 73 57 65)(50 72 58 80)(51 71 59 79)(52 70 60 78)(53 69 61 77)(54 68 62 76)(55 67 63 75)(56 66 64 74)(81 105 89 97)(82 104 90 112)(83 103 91 111)(84 102 92 110)(85 101 93 109)(86 100 94 108)(87 99 95 107)(88 98 96 106)
G:=sub<Sym(128)| (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (1,92)(2,93)(3,94)(4,95)(5,96)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,97)(32,98)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,126)(50,127)(51,128)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125), (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), (1,28,9,20)(2,27,10,19)(3,26,11,18)(4,25,12,17)(5,24,13,32)(6,23,14,31)(7,22,15,30)(8,21,16,29)(33,118,41,126)(34,117,42,125)(35,116,43,124)(36,115,44,123)(37,114,45,122)(38,113,46,121)(39,128,47,120)(40,127,48,119)(49,73,57,65)(50,72,58,80)(51,71,59,79)(52,70,60,78)(53,69,61,77)(54,68,62,76)(55,67,63,75)(56,66,64,74)(81,105,89,97)(82,104,90,112)(83,103,91,111)(84,102,92,110)(85,101,93,109)(86,100,94,108)(87,99,95,107)(88,98,96,106)>;
G:=Group( (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (1,92)(2,93)(3,94)(4,95)(5,96)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,97)(32,98)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,126)(50,127)(51,128)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125), (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), (1,28,9,20)(2,27,10,19)(3,26,11,18)(4,25,12,17)(5,24,13,32)(6,23,14,31)(7,22,15,30)(8,21,16,29)(33,118,41,126)(34,117,42,125)(35,116,43,124)(36,115,44,123)(37,114,45,122)(38,113,46,121)(39,128,47,120)(40,127,48,119)(49,73,57,65)(50,72,58,80)(51,71,59,79)(52,70,60,78)(53,69,61,77)(54,68,62,76)(55,67,63,75)(56,66,64,74)(81,105,89,97)(82,104,90,112)(83,103,91,111)(84,102,92,110)(85,101,93,109)(86,100,94,108)(87,99,95,107)(88,98,96,106) );
G=PermutationGroup([[(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,97),(32,98),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,126),(50,127),(51,128),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125)], [(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)], [(1,28,9,20),(2,27,10,19),(3,26,11,18),(4,25,12,17),(5,24,13,32),(6,23,14,31),(7,22,15,30),(8,21,16,29),(33,118,41,126),(34,117,42,125),(35,116,43,124),(36,115,44,123),(37,114,45,122),(38,113,46,121),(39,128,47,120),(40,127,48,119),(49,73,57,65),(50,72,58,80),(51,71,59,79),(52,70,60,78),(53,69,61,77),(54,68,62,76),(55,67,63,75),(56,66,64,74),(81,105,89,97),(82,104,90,112),(83,103,91,111),(84,102,92,110),(85,101,93,109),(86,100,94,108),(87,99,95,107),(88,98,96,106)]])
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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | Q32 |
| kernel | C22×Q32 | C22×C16 | C2×Q32 | C22×Q16 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 3 | 1 | 6 | 2 | 16 |
Matrix representation of C22×Q32 ►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 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 11 | 0 | 0 |
| 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 12 |
| 0 | 0 | 0 | 11 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 12 | 10 |
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],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,6,0,0,0,11,1,0,0,0,0,0,7,11,0,0,0,12,2],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,7,12,0,0,0,10,10] >;
C22×Q32 in GAP, Magma, Sage, TeX
C_2^2\times Q_{32} % in TeX
G:=Group("C2^2xQ32"); // GroupNames label
G:=SmallGroup(128,2142);
// by ID
G=gap.SmallGroup(128,2142);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,456,1684,851,242,4037,2028,124]);
// 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=c^-1>;
// generators/relations