direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23×Q16, C4.3C25, C8.19C24, Q8.1C24, C24.197D4, (C23×C8).16C2, C4.29(C22×D4), C2.38(D4×C23), (C2×C8).574C23, (C2×C4).609C24, C23.895(C2×D4), (C22×C4).629D4, (Q8×C23).15C2, (C2×Q8).473C23, (C23×C4).713C22, (C22×C8).544C22, C22.166(C22×D4), (C22×C4).1591C23, (C22×Q8).503C22, (C2×C4).882(C2×D4), SmallGroup(128,2308)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 988 in 732 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C4 [×16], C22 [×35], C8 [×8], C2×C4 [×28], C2×C4 [×56], Q8 [×16], Q8 [×56], C23 [×15], C2×C8 [×28], Q16 [×64], C22×C4 [×14], C22×C4 [×28], C2×Q8 [×56], C2×Q8 [×84], C24, C22×C8 [×14], C2×Q16 [×112], C23×C4, C23×C4 [×2], C22×Q8 [×28], C22×Q8 [×14], C23×C8, C22×Q16 [×28], Q8×C23 [×2], C23×Q16
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], Q16 [×8], C2×D4 [×28], C24 [×31], C2×Q16 [×28], C22×D4 [×14], C25, C22×Q16 [×14], D4×C23, C23×Q16
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►Regular action on 128 points
Generators in S128
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 97)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 65)(31 66)(32 67)(33 125)(34 126)(35 127)(36 128)(37 121)(38 122)(39 123)(40 124)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 113)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 89)(81 109)(82 110)(83 111)(84 112)(85 105)(86 106)(87 107)(88 108)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(41 69)(42 70)(43 71)(44 72)(45 65)(46 66)(47 67)(48 68)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 81)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)(97 126)(98 127)(99 128)(100 121)(101 122)(102 123)(103 124)(104 125)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 89)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 105)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 121)
(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 62 5 58)(2 61 6 57)(3 60 7 64)(4 59 8 63)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 29 21 25)(18 28 22 32)(19 27 23 31)(20 26 24 30)(33 76 37 80)(34 75 38 79)(35 74 39 78)(36 73 40 77)(41 84 45 88)(42 83 46 87)(43 82 47 86)(44 81 48 85)(49 67 53 71)(50 66 54 70)(51 65 55 69)(52 72 56 68)(89 125 93 121)(90 124 94 128)(91 123 95 127)(92 122 96 126)(105 117 109 113)(106 116 110 120)(107 115 111 119)(108 114 112 118)
G:=sub<Sym(128)| (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,97)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,68)(26,69)(27,70)(28,71)(29,72)(30,65)(31,66)(32,67)(33,125)(34,126)(35,127)(36,128)(37,121)(38,122)(39,123)(40,124)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,113)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89)(81,109)(82,110)(83,111)(84,112)(85,105)(86,106)(87,107)(88,108), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,81)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76)(97,126)(98,127)(99,128)(100,121)(101,122)(102,123)(103,124)(104,125), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,89)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,105)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121), (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), (1,62,5,58)(2,61,6,57)(3,60,7,64)(4,59,8,63)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30)(33,76,37,80)(34,75,38,79)(35,74,39,78)(36,73,40,77)(41,84,45,88)(42,83,46,87)(43,82,47,86)(44,81,48,85)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(89,125,93,121)(90,124,94,128)(91,123,95,127)(92,122,96,126)(105,117,109,113)(106,116,110,120)(107,115,111,119)(108,114,112,118)>;
G:=Group( (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,97)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,68)(26,69)(27,70)(28,71)(29,72)(30,65)(31,66)(32,67)(33,125)(34,126)(35,127)(36,128)(37,121)(38,122)(39,123)(40,124)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,113)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89)(81,109)(82,110)(83,111)(84,112)(85,105)(86,106)(87,107)(88,108), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,81)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76)(97,126)(98,127)(99,128)(100,121)(101,122)(102,123)(103,124)(104,125), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,89)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,105)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121), (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), (1,62,5,58)(2,61,6,57)(3,60,7,64)(4,59,8,63)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30)(33,76,37,80)(34,75,38,79)(35,74,39,78)(36,73,40,77)(41,84,45,88)(42,83,46,87)(43,82,47,86)(44,81,48,85)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(89,125,93,121)(90,124,94,128)(91,123,95,127)(92,122,96,126)(105,117,109,113)(106,116,110,120)(107,115,111,119)(108,114,112,118) );
G=PermutationGroup([(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,97),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(25,68),(26,69),(27,70),(28,71),(29,72),(30,65),(31,66),(32,67),(33,125),(34,126),(35,127),(36,128),(37,121),(38,122),(39,123),(40,124),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,113),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,89),(81,109),(82,110),(83,111),(84,112),(85,105),(86,106),(87,107),(88,108)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(41,69),(42,70),(43,71),(44,72),(45,65),(46,66),(47,67),(48,68),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,81),(57,77),(58,78),(59,79),(60,80),(61,73),(62,74),(63,75),(64,76),(97,126),(98,127),(99,128),(100,121),(101,122),(102,123),(103,124),(104,125)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,113),(8,114),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,76),(18,77),(19,78),(20,79),(21,80),(22,73),(23,74),(24,75),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,89),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,105),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,121)], [(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)], [(1,62,5,58),(2,61,6,57),(3,60,7,64),(4,59,8,63),(9,104,13,100),(10,103,14,99),(11,102,15,98),(12,101,16,97),(17,29,21,25),(18,28,22,32),(19,27,23,31),(20,26,24,30),(33,76,37,80),(34,75,38,79),(35,74,39,78),(36,73,40,77),(41,84,45,88),(42,83,46,87),(43,82,47,86),(44,81,48,85),(49,67,53,71),(50,66,54,70),(51,65,55,69),(52,72,56,68),(89,125,93,121),(90,124,94,128),(91,123,95,127),(92,122,96,126),(105,117,109,113),(106,116,110,120),(107,115,111,119),(108,114,112,118)])
Matrix representation ►G ⊆ GL7(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 | 7 |
G:=sub<GL(7,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,7,4,0,0,0,0,0,13,10,0,0,0,0,0,0,0,10,13,0,0,0,0,0,4,7,0,0,0,0,0,0,0,3,3,0,0,0,0,0,14,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,16,7] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D4 | Q16 |
| kernel | C23×Q16 | C23×C8 | C22×Q16 | Q8×C23 | C22×C4 | C24 | C23 |
| # reps | 1 | 1 | 28 | 2 | 7 | 1 | 16 |
In GAP, Magma, Sage, TeX
C_2^3\times Q_{16} % in TeX
G:=Group("C2^3xQ16"); // GroupNames label
G:=SmallGroup(128,2308);
// by ID
G=gap.SmallGroup(128,2308);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀