direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×C2.D8, C23.62D8, C24.192D4, C23.24Q16, C8⋊8(C22×C4), (C22×C8)⋊15C4, C2.2(C22×D8), C4.2(C22×Q8), C4.46(C23×C4), (C23×C8).14C2, C22.68(C2×D8), C2.2(C22×Q16), C4⋊C4.347C23, C23.86(C4⋊C4), (C2×C8).554C23, (C2×C4).184C24, (C22×C4).604D4, C23.839(C2×D4), C22.45(C2×Q16), (C22×C4).102Q8, (C22×C8).533C22, (C23×C4).695C22, C22.131(C22×D4), (C22×C4).1505C23, (C2×C8)⋊36(C2×C4), C4.64(C2×C4⋊C4), (C2×C4).842(C2×D4), C22.75(C2×C4⋊C4), C2.23(C22×C4⋊C4), (C2×C4).238(C2×Q8), (C2×C4).150(C4⋊C4), (C22×C4⋊C4).43C2, (C2×C4⋊C4).902C22, (C2×C4).572(C22×C4), (C22×C4).495(C2×C4), SmallGroup(128,1640)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 460 in 300 conjugacy classes, 220 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×27], C2×C4 [×32], C23 [×15], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×28], C22×C4 [×14], C22×C4 [×20], C24, C2.D8 [×16], C2×C4⋊C4 [×12], C2×C4⋊C4 [×6], C22×C8 [×14], C23×C4, C23×C4 [×2], C2×C2.D8 [×12], C22×C4⋊C4 [×2], C23×C8, C22×C2.D8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], D8 [×4], Q16 [×4], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2.D8 [×16], C2×C4⋊C4 [×12], C2×D8 [×6], C2×Q16 [×6], C23×C4, C22×D4, C22×Q8, C2×C2.D8 [×12], C22×C4⋊C4, C22×D8, C22×Q16, C22×C2.D8
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, 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 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(65 124)(66 125)(67 126)(68 127)(69 128)(70 121)(71 122)(72 123)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 89)(80 90)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 124)(50 125)(51 126)(52 127)(53 128)(54 121)(55 122)(56 123)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 104)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(89 127)(90 128)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(105 119)(106 120)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)
(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)
(1 116 59 110)(2 115 60 109)(3 114 61 108)(4 113 62 107)(5 120 63 106)(6 119 64 105)(7 118 57 112)(8 117 58 111)(9 103 34 51)(10 102 35 50)(11 101 36 49)(12 100 37 56)(13 99 38 55)(14 98 39 54)(15 97 40 53)(16 104 33 52)(17 76 83 65)(18 75 84 72)(19 74 85 71)(20 73 86 70)(21 80 87 69)(22 79 88 68)(23 78 81 67)(24 77 82 66)(25 128 47 90)(26 127 48 89)(27 126 41 96)(28 125 42 95)(29 124 43 94)(30 123 44 93)(31 122 45 92)(32 121 46 91)
G:=sub<Sym(128)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(65,124)(66,125)(67,126)(68,127)(69,128)(70,121)(71,122)(72,123)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90)(97,116)(98,117)(99,118)(100,119)(101,120)(102,113)(103,114)(104,115), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,124)(50,125)(51,126)(52,127)(53,128)(54,121)(55,122)(56,123)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,104)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(89,127)(90,128)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(105,119)(106,120)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (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), (1,116,59,110)(2,115,60,109)(3,114,61,108)(4,113,62,107)(5,120,63,106)(6,119,64,105)(7,118,57,112)(8,117,58,111)(9,103,34,51)(10,102,35,50)(11,101,36,49)(12,100,37,56)(13,99,38,55)(14,98,39,54)(15,97,40,53)(16,104,33,52)(17,76,83,65)(18,75,84,72)(19,74,85,71)(20,73,86,70)(21,80,87,69)(22,79,88,68)(23,78,81,67)(24,77,82,66)(25,128,47,90)(26,127,48,89)(27,126,41,96)(28,125,42,95)(29,124,43,94)(30,123,44,93)(31,122,45,92)(32,121,46,91)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(65,124)(66,125)(67,126)(68,127)(69,128)(70,121)(71,122)(72,123)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90)(97,116)(98,117)(99,118)(100,119)(101,120)(102,113)(103,114)(104,115), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,124)(50,125)(51,126)(52,127)(53,128)(54,121)(55,122)(56,123)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(73,117)(74,118)(75,119)(76,120)(77,113)(78,114)(79,115)(80,116)(89,104)(90,97)(91,98)(92,99)(93,100)(94,101)(95,102)(96,103), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(65,76)(66,77)(67,78)(68,79)(69,80)(70,73)(71,74)(72,75)(89,127)(90,128)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(105,119)(106,120)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (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), (1,116,59,110)(2,115,60,109)(3,114,61,108)(4,113,62,107)(5,120,63,106)(6,119,64,105)(7,118,57,112)(8,117,58,111)(9,103,34,51)(10,102,35,50)(11,101,36,49)(12,100,37,56)(13,99,38,55)(14,98,39,54)(15,97,40,53)(16,104,33,52)(17,76,83,65)(18,75,84,72)(19,74,85,71)(20,73,86,70)(21,80,87,69)(22,79,88,68)(23,78,81,67)(24,77,82,66)(25,128,47,90)(26,127,48,89)(27,126,41,96)(28,125,42,95)(29,124,43,94)(30,123,44,93)(31,122,45,92)(32,121,46,91) );
G=PermutationGroup([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,87),(26,88),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,105),(65,124),(66,125),(67,126),(68,127),(69,128),(70,121),(71,122),(72,123),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,89),(80,90),(97,116),(98,117),(99,118),(100,119),(101,120),(102,113),(103,114),(104,115)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,124),(50,125),(51,126),(52,127),(53,128),(54,121),(55,122),(56,123),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(73,117),(74,118),(75,119),(76,120),(77,113),(78,114),(79,115),(80,116),(89,104),(90,97),(91,98),(92,99),(93,100),(94,101),(95,102),(96,103)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,81),(24,82),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(49,101),(50,102),(51,103),(52,104),(53,97),(54,98),(55,99),(56,100),(65,76),(66,77),(67,78),(68,79),(69,80),(70,73),(71,74),(72,75),(89,127),(90,128),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(105,119),(106,120),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118)], [(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)], [(1,116,59,110),(2,115,60,109),(3,114,61,108),(4,113,62,107),(5,120,63,106),(6,119,64,105),(7,118,57,112),(8,117,58,111),(9,103,34,51),(10,102,35,50),(11,101,36,49),(12,100,37,56),(13,99,38,55),(14,98,39,54),(15,97,40,53),(16,104,33,52),(17,76,83,65),(18,75,84,72),(19,74,85,71),(20,73,86,70),(21,80,87,69),(22,79,88,68),(23,78,81,67),(24,77,82,66),(25,128,47,90),(26,127,48,89),(27,126,41,96),(28,125,42,95),(29,124,43,94),(30,123,44,93),(31,122,45,92),(32,121,46,91)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 16 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,8,0,0,0,0,0,15],[16,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,0,9,0,0,0,15,0] >;
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 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 |
| kernel | C22×C2.D8 | C2×C2.D8 | C22×C4⋊C4 | C23×C8 | C22×C8 | C22×C4 | C22×C4 | C24 | C23 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 3 | 4 | 1 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_2.D_8
% in TeX
G:=Group("C2^2xC2.D8"); // GroupNames label
G:=SmallGroup(128,1640);
// by ID
G=gap.SmallGroup(128,1640);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,568,2804,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,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
○
≀