direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22×Q32, C8.11C24, C23.65D8, C16.12C23, Q16.1C23, (C2×C4).96D8, C4.23(C2×D8), C8.56(C2×D4), (C2×C8).264D4, C22.77(C2×D8), C2.26(C22×D8), C4.17(C22×D4), (C2×C8).573C23, (C22×C16).12C2, (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
Subgroups: 340 in 180 conjugacy classes, 100 normal (9 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C8, C8 [×3], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, C16 [×4], C2×C8 [×6], Q16 [×8], Q16 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×18], C2×C16 [×6], Q32 [×16], C22×C8, C2×Q16 [×12], C2×Q16 [×6], C22×Q8 [×2], C22×C16, C2×Q32 [×12], C22×Q16 [×2], C22×Q32
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, Q32 [×4], C2×D8 [×6], C22×D4, C2×Q32 [×6], C22×D8, C22×Q32
Generators and relations
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 >
►Regular action on 128 points
Generators in S128
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 49)(13 50)(14 51)(15 52)(16 53)(17 46)(18 47)(19 48)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 128)(110 113)(111 114)(112 115)
(1 96)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 125)(34 126)(35 127)(36 128)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 65)(63 66)(64 67)
(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)
(1 118 9 126)(2 117 10 125)(3 116 11 124)(4 115 12 123)(5 114 13 122)(6 113 14 121)(7 128 15 120)(8 127 16 119)(17 77 25 69)(18 76 26 68)(19 75 27 67)(20 74 28 66)(21 73 29 65)(22 72 30 80)(23 71 31 79)(24 70 32 78)(33 81 41 89)(34 96 42 88)(35 95 43 87)(36 94 44 86)(37 93 45 85)(38 92 46 84)(39 91 47 83)(40 90 48 82)(49 104 57 112)(50 103 58 111)(51 102 59 110)(52 101 60 109)(53 100 61 108)(54 99 62 107)(55 98 63 106)(56 97 64 105)
G:=sub<Sym(128)| (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,49)(13,50)(14,51)(15,52)(16,53)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,113)(111,114)(112,115), (1,96)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,65)(63,66)(64,67), (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), (1,118,9,126)(2,117,10,125)(3,116,11,124)(4,115,12,123)(5,114,13,122)(6,113,14,121)(7,128,15,120)(8,127,16,119)(17,77,25,69)(18,76,26,68)(19,75,27,67)(20,74,28,66)(21,73,29,65)(22,72,30,80)(23,71,31,79)(24,70,32,78)(33,81,41,89)(34,96,42,88)(35,95,43,87)(36,94,44,86)(37,93,45,85)(38,92,46,84)(39,91,47,83)(40,90,48,82)(49,104,57,112)(50,103,58,111)(51,102,59,110)(52,101,60,109)(53,100,61,108)(54,99,62,107)(55,98,63,106)(56,97,64,105)>;
G:=Group( (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,49)(13,50)(14,51)(15,52)(16,53)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,113)(111,114)(112,115), (1,96)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,125)(34,126)(35,127)(36,128)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,65)(63,66)(64,67), (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), (1,118,9,126)(2,117,10,125)(3,116,11,124)(4,115,12,123)(5,114,13,122)(6,113,14,121)(7,128,15,120)(8,127,16,119)(17,77,25,69)(18,76,26,68)(19,75,27,67)(20,74,28,66)(21,73,29,65)(22,72,30,80)(23,71,31,79)(24,70,32,78)(33,81,41,89)(34,96,42,88)(35,95,43,87)(36,94,44,86)(37,93,45,85)(38,92,46,84)(39,91,47,83)(40,90,48,82)(49,104,57,112)(50,103,58,111)(51,102,59,110)(52,101,60,109)(53,100,61,108)(54,99,62,107)(55,98,63,106)(56,97,64,105) );
G=PermutationGroup([(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,49),(13,50),(14,51),(15,52),(16,53),(17,46),(18,47),(19,48),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124),(106,125),(107,126),(108,127),(109,128),(110,113),(111,114),(112,115)], [(1,96),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,125),(34,126),(35,127),(36,128),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,65),(63,66),(64,67)], [(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)], [(1,118,9,126),(2,117,10,125),(3,116,11,124),(4,115,12,123),(5,114,13,122),(6,113,14,121),(7,128,15,120),(8,127,16,119),(17,77,25,69),(18,76,26,68),(19,75,27,67),(20,74,28,66),(21,73,29,65),(22,72,30,80),(23,71,31,79),(24,70,32,78),(33,81,41,89),(34,96,42,88),(35,95,43,87),(36,94,44,86),(37,93,45,85),(38,92,46,84),(39,91,47,83),(40,90,48,82),(49,104,57,112),(50,103,58,111),(51,102,59,110),(52,101,60,109),(53,100,61,108),(54,99,62,107),(55,98,63,106),(56,97,64,105)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
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