direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C22×C4, C22.9C25, C24.655C23, C23.267C24, C42.736C23, C2.5(C24×C4), C2.2(Q8×C23), C4.35(C23×C4), C4⋊C4.511C23, (C2×C4).156C24, (Q8×C23).16C2, C23.148(C2×Q8), (C22×C42).35C2, C22.48(C23×C4), (C2×Q8).477C23, C23.376(C4○D4), C22.49(C22×Q8), C23.296(C22×C4), (C23×C4).720C22, (C2×C42).1138C22, (C22×C4).1582C23, (C22×Q8).511C22, C4○(C2×C4×Q8), (C2×C4)○2(C4×Q8), C4⋊C4○4(C22×C4), C4○3(C22×C4⋊C4), (C22×C4)○(C4×Q8), C2.3(C22×C4○D4), (C22×C4⋊C4).51C2, (C2×C4⋊C4).980C22, (C2×C4).478(C22×C4), (C22×C4).422(C2×C4), C22.144(C2×C4○D4), (C2×C4)○(C2×C4×Q8), (C2×C4)○6(C2×C4⋊C4), (C22×C4)○(C2×C4×Q8), (C2×C4)○3(C22×C4⋊C4), (C22×C4)○4(C2×C4⋊C4), (C22×C4)○3(C22×C4⋊C4), SmallGroup(128,2155)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 892 in 832 conjugacy classes, 772 normal (9 characteristic)
C1, C2 [×3], C2 [×12], C4 [×32], C4 [×12], C22, C22 [×34], C2×C4 [×124], C2×C4 [×36], Q8 [×64], C23 [×15], C42 [×48], C4⋊C4 [×48], C22×C4 [×74], C22×C4 [×12], C2×Q8 [×112], C24, C2×C42 [×36], C2×C4⋊C4 [×36], C4×Q8 [×64], C23×C4, C23×C4 [×6], C22×Q8 [×28], C22×C42 [×3], C22×C4⋊C4 [×3], C2×C4×Q8 [×24], Q8×C23, Q8×C22×C4
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], Q8 [×8], C23 [×155], C22×C4 [×140], C2×Q8 [×28], C4○D4 [×4], C24 [×31], C4×Q8 [×16], C23×C4 [×30], C22×Q8 [×14], C2×C4○D4 [×6], C25, C2×C4×Q8 [×12], C24×C4, Q8×C23, C22×C4○D4, Q8×C22×C4
Generators and relations
G = < a,b,c,d,e | a2=b2=c4=d4=1, e2=d2, 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 21)(2 22)(3 23)(4 24)(5 43)(6 44)(7 41)(8 42)(9 15)(10 16)(11 13)(12 14)(17 32)(18 29)(19 30)(20 31)(25 103)(26 104)(27 101)(28 102)(33 47)(34 48)(35 45)(36 46)(37 75)(38 76)(39 73)(40 74)(49 53)(50 54)(51 55)(52 56)(57 63)(58 64)(59 61)(60 62)(65 79)(66 80)(67 77)(68 78)(69 107)(70 108)(71 105)(72 106)(81 85)(82 86)(83 87)(84 88)(89 95)(90 96)(91 93)(92 94)(97 111)(98 112)(99 109)(100 110)(113 117)(114 118)(115 119)(116 120)(121 127)(122 128)(123 125)(124 126)
(1 43)(2 44)(3 41)(4 42)(5 21)(6 22)(7 23)(8 24)(9 29)(10 30)(11 31)(12 32)(13 20)(14 17)(15 18)(16 19)(25 125)(26 126)(27 127)(28 128)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)(61 73)(62 74)(63 75)(64 76)(65 88)(66 85)(67 86)(68 87)(69 89)(70 90)(71 91)(72 92)(77 82)(78 83)(79 84)(80 81)(93 105)(94 106)(95 107)(96 108)(97 120)(98 117)(99 118)(100 119)(101 121)(102 122)(103 123)(104 124)(109 114)(110 115)(111 116)(112 113)
(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 57 9 35)(2 58 10 36)(3 59 11 33)(4 60 12 34)(5 75 18 50)(6 76 19 51)(7 73 20 52)(8 74 17 49)(13 47 23 61)(14 48 24 62)(15 45 21 63)(16 46 22 64)(25 71 116 88)(26 72 113 85)(27 69 114 86)(28 70 115 87)(29 54 43 37)(30 55 44 38)(31 56 41 39)(32 53 42 40)(65 125 91 111)(66 126 92 112)(67 127 89 109)(68 128 90 110)(77 121 95 99)(78 122 96 100)(79 123 93 97)(80 124 94 98)(81 104 106 117)(82 101 107 118)(83 102 108 119)(84 103 105 120)
(1 72 9 85)(2 69 10 86)(3 70 11 87)(4 71 12 88)(5 94 18 80)(6 95 19 77)(7 96 20 78)(8 93 17 79)(13 83 23 108)(14 84 24 105)(15 81 21 106)(16 82 22 107)(25 34 116 60)(26 35 113 57)(27 36 114 58)(28 33 115 59)(29 66 43 92)(30 67 44 89)(31 68 41 90)(32 65 42 91)(37 126 54 112)(38 127 55 109)(39 128 56 110)(40 125 53 111)(45 117 63 104)(46 118 64 101)(47 119 61 102)(48 120 62 103)(49 97 74 123)(50 98 75 124)(51 99 76 121)(52 100 73 122)
G:=sub<Sym(128)| (1,21)(2,22)(3,23)(4,24)(5,43)(6,44)(7,41)(8,42)(9,15)(10,16)(11,13)(12,14)(17,32)(18,29)(19,30)(20,31)(25,103)(26,104)(27,101)(28,102)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(49,53)(50,54)(51,55)(52,56)(57,63)(58,64)(59,61)(60,62)(65,79)(66,80)(67,77)(68,78)(69,107)(70,108)(71,105)(72,106)(81,85)(82,86)(83,87)(84,88)(89,95)(90,96)(91,93)(92,94)(97,111)(98,112)(99,109)(100,110)(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)(124,126), (1,43)(2,44)(3,41)(4,42)(5,21)(6,22)(7,23)(8,24)(9,29)(10,30)(11,31)(12,32)(13,20)(14,17)(15,18)(16,19)(25,125)(26,126)(27,127)(28,128)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49)(61,73)(62,74)(63,75)(64,76)(65,88)(66,85)(67,86)(68,87)(69,89)(70,90)(71,91)(72,92)(77,82)(78,83)(79,84)(80,81)(93,105)(94,106)(95,107)(96,108)(97,120)(98,117)(99,118)(100,119)(101,121)(102,122)(103,123)(104,124)(109,114)(110,115)(111,116)(112,113), (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), (1,57,9,35)(2,58,10,36)(3,59,11,33)(4,60,12,34)(5,75,18,50)(6,76,19,51)(7,73,20,52)(8,74,17,49)(13,47,23,61)(14,48,24,62)(15,45,21,63)(16,46,22,64)(25,71,116,88)(26,72,113,85)(27,69,114,86)(28,70,115,87)(29,54,43,37)(30,55,44,38)(31,56,41,39)(32,53,42,40)(65,125,91,111)(66,126,92,112)(67,127,89,109)(68,128,90,110)(77,121,95,99)(78,122,96,100)(79,123,93,97)(80,124,94,98)(81,104,106,117)(82,101,107,118)(83,102,108,119)(84,103,105,120), (1,72,9,85)(2,69,10,86)(3,70,11,87)(4,71,12,88)(5,94,18,80)(6,95,19,77)(7,96,20,78)(8,93,17,79)(13,83,23,108)(14,84,24,105)(15,81,21,106)(16,82,22,107)(25,34,116,60)(26,35,113,57)(27,36,114,58)(28,33,115,59)(29,66,43,92)(30,67,44,89)(31,68,41,90)(32,65,42,91)(37,126,54,112)(38,127,55,109)(39,128,56,110)(40,125,53,111)(45,117,63,104)(46,118,64,101)(47,119,61,102)(48,120,62,103)(49,97,74,123)(50,98,75,124)(51,99,76,121)(52,100,73,122)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,43)(6,44)(7,41)(8,42)(9,15)(10,16)(11,13)(12,14)(17,32)(18,29)(19,30)(20,31)(25,103)(26,104)(27,101)(28,102)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(49,53)(50,54)(51,55)(52,56)(57,63)(58,64)(59,61)(60,62)(65,79)(66,80)(67,77)(68,78)(69,107)(70,108)(71,105)(72,106)(81,85)(82,86)(83,87)(84,88)(89,95)(90,96)(91,93)(92,94)(97,111)(98,112)(99,109)(100,110)(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)(124,126), (1,43)(2,44)(3,41)(4,42)(5,21)(6,22)(7,23)(8,24)(9,29)(10,30)(11,31)(12,32)(13,20)(14,17)(15,18)(16,19)(25,125)(26,126)(27,127)(28,128)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49)(61,73)(62,74)(63,75)(64,76)(65,88)(66,85)(67,86)(68,87)(69,89)(70,90)(71,91)(72,92)(77,82)(78,83)(79,84)(80,81)(93,105)(94,106)(95,107)(96,108)(97,120)(98,117)(99,118)(100,119)(101,121)(102,122)(103,123)(104,124)(109,114)(110,115)(111,116)(112,113), (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), (1,57,9,35)(2,58,10,36)(3,59,11,33)(4,60,12,34)(5,75,18,50)(6,76,19,51)(7,73,20,52)(8,74,17,49)(13,47,23,61)(14,48,24,62)(15,45,21,63)(16,46,22,64)(25,71,116,88)(26,72,113,85)(27,69,114,86)(28,70,115,87)(29,54,43,37)(30,55,44,38)(31,56,41,39)(32,53,42,40)(65,125,91,111)(66,126,92,112)(67,127,89,109)(68,128,90,110)(77,121,95,99)(78,122,96,100)(79,123,93,97)(80,124,94,98)(81,104,106,117)(82,101,107,118)(83,102,108,119)(84,103,105,120), (1,72,9,85)(2,69,10,86)(3,70,11,87)(4,71,12,88)(5,94,18,80)(6,95,19,77)(7,96,20,78)(8,93,17,79)(13,83,23,108)(14,84,24,105)(15,81,21,106)(16,82,22,107)(25,34,116,60)(26,35,113,57)(27,36,114,58)(28,33,115,59)(29,66,43,92)(30,67,44,89)(31,68,41,90)(32,65,42,91)(37,126,54,112)(38,127,55,109)(39,128,56,110)(40,125,53,111)(45,117,63,104)(46,118,64,101)(47,119,61,102)(48,120,62,103)(49,97,74,123)(50,98,75,124)(51,99,76,121)(52,100,73,122) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,43),(6,44),(7,41),(8,42),(9,15),(10,16),(11,13),(12,14),(17,32),(18,29),(19,30),(20,31),(25,103),(26,104),(27,101),(28,102),(33,47),(34,48),(35,45),(36,46),(37,75),(38,76),(39,73),(40,74),(49,53),(50,54),(51,55),(52,56),(57,63),(58,64),(59,61),(60,62),(65,79),(66,80),(67,77),(68,78),(69,107),(70,108),(71,105),(72,106),(81,85),(82,86),(83,87),(84,88),(89,95),(90,96),(91,93),(92,94),(97,111),(98,112),(99,109),(100,110),(113,117),(114,118),(115,119),(116,120),(121,127),(122,128),(123,125),(124,126)], [(1,43),(2,44),(3,41),(4,42),(5,21),(6,22),(7,23),(8,24),(9,29),(10,30),(11,31),(12,32),(13,20),(14,17),(15,18),(16,19),(25,125),(26,126),(27,127),(28,128),(33,56),(34,53),(35,54),(36,55),(37,57),(38,58),(39,59),(40,60),(45,50),(46,51),(47,52),(48,49),(61,73),(62,74),(63,75),(64,76),(65,88),(66,85),(67,86),(68,87),(69,89),(70,90),(71,91),(72,92),(77,82),(78,83),(79,84),(80,81),(93,105),(94,106),(95,107),(96,108),(97,120),(98,117),(99,118),(100,119),(101,121),(102,122),(103,123),(104,124),(109,114),(110,115),(111,116),(112,113)], [(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)], [(1,57,9,35),(2,58,10,36),(3,59,11,33),(4,60,12,34),(5,75,18,50),(6,76,19,51),(7,73,20,52),(8,74,17,49),(13,47,23,61),(14,48,24,62),(15,45,21,63),(16,46,22,64),(25,71,116,88),(26,72,113,85),(27,69,114,86),(28,70,115,87),(29,54,43,37),(30,55,44,38),(31,56,41,39),(32,53,42,40),(65,125,91,111),(66,126,92,112),(67,127,89,109),(68,128,90,110),(77,121,95,99),(78,122,96,100),(79,123,93,97),(80,124,94,98),(81,104,106,117),(82,101,107,118),(83,102,108,119),(84,103,105,120)], [(1,72,9,85),(2,69,10,86),(3,70,11,87),(4,71,12,88),(5,94,18,80),(6,95,19,77),(7,96,20,78),(8,93,17,79),(13,83,23,108),(14,84,24,105),(15,81,21,106),(16,82,22,107),(25,34,116,60),(26,35,113,57),(27,36,114,58),(28,33,115,59),(29,66,43,92),(30,67,44,89),(31,68,41,90),(32,65,42,91),(37,126,54,112),(38,127,55,109),(39,128,56,110),(40,125,53,111),(45,117,63,104),(46,118,64,101),(47,119,61,102),(48,120,62,103),(49,97,74,123),(50,98,75,124),(51,99,76,121),(52,100,73,122)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 2 | 4 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,2],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,2,0,0,0,4,4] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 |
| kernel | Q8×C22×C4 | C22×C42 | C22×C4⋊C4 | C2×C4×Q8 | Q8×C23 | C22×Q8 | C22×C4 | C23 |
| # reps | 1 | 3 | 3 | 24 | 1 | 32 | 8 | 8 |
In GAP, Magma, Sage, TeX
Q_8\times C_2^2\times C_4
% in TeX
G:=Group("Q8xC2^2xC4"); // GroupNames label
G:=SmallGroup(128,2155);
// by ID
G=gap.SmallGroup(128,2155);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,520]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^4=1,e^2=d^2,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
○
≀