direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊3Q8, C42.361D4, C42.715C23, C8⋊6(C2×Q8), (C2×C8)⋊14Q8, C4.12(C4⋊Q8), C4.8(C22×Q8), C4⋊C4.96C23, C4.22(C2×SD16), (C2×C4).81SD16, (C2×C8).596C23, (C4×C8).430C22, (C2×C4).355C24, C23.882(C2×D4), (C22×C4).615D4, C4⋊Q8.282C22, C22.44(C4⋊Q8), C22.89(C2×SD16), C2.19(C22×SD16), C4.Q8.157C22, (C22×C8).567C22, C22.615(C22×D4), (C22×C4).1564C23, (C2×C42).1130C22, (C2×C4×C8).56C2, C2.25(C2×C4⋊Q8), (C2×C4⋊Q8).49C2, (C2×C4).860(C2×D4), (C2×C4).244(C2×Q8), (C2×C4.Q8).33C2, (C2×C4⋊C4).628C22, SmallGroup(128,1889)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 212 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×16], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], C4×C8 [×4], C4.Q8 [×16], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C2×C4×C8, C2×C4.Q8 [×4], C8⋊3Q8 [×8], C2×C4⋊Q8 [×2], C2×C8⋊3Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], SD16 [×8], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C2×SD16 [×12], C22×D4, C22×Q8 [×2], C8⋊3Q8 [×4], C2×C4⋊Q8, C22×SD16 [×2], C2×C8⋊3Q8
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >
►Regular action on 128 points
Generators in S128
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 104)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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 63 47 38)(2 64 48 39)(3 57 41 40)(4 58 42 33)(5 59 43 34)(6 60 44 35)(7 61 45 36)(8 62 46 37)(9 80 18 53)(10 73 19 54)(11 74 20 55)(12 75 21 56)(13 76 22 49)(14 77 23 50)(15 78 24 51)(16 79 17 52)(25 68 114 93)(26 69 115 94)(27 70 116 95)(28 71 117 96)(29 72 118 89)(30 65 119 90)(31 66 120 91)(32 67 113 92)(81 99 106 124)(82 100 107 125)(83 101 108 126)(84 102 109 127)(85 103 110 128)(86 104 111 121)(87 97 112 122)(88 98 105 123)
(1 93 47 68)(2 96 48 71)(3 91 41 66)(4 94 42 69)(5 89 43 72)(6 92 44 67)(7 95 45 70)(8 90 46 65)(9 108 18 83)(10 111 19 86)(11 106 20 81)(12 109 21 84)(13 112 22 87)(14 107 23 82)(15 110 24 85)(16 105 17 88)(25 63 114 38)(26 58 115 33)(27 61 116 36)(28 64 117 39)(29 59 118 34)(30 62 119 37)(31 57 120 40)(32 60 113 35)(49 122 76 97)(50 125 77 100)(51 128 78 103)(52 123 79 98)(53 126 80 101)(54 121 73 104)(55 124 74 99)(56 127 75 102)
G:=sub<Sym(128)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,104)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,76)(34,77)(35,78)(36,79)(37,80)(38,73)(39,74)(40,75)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,63,47,38)(2,64,48,39)(3,57,41,40)(4,58,42,33)(5,59,43,34)(6,60,44,35)(7,61,45,36)(8,62,46,37)(9,80,18,53)(10,73,19,54)(11,74,20,55)(12,75,21,56)(13,76,22,49)(14,77,23,50)(15,78,24,51)(16,79,17,52)(25,68,114,93)(26,69,115,94)(27,70,116,95)(28,71,117,96)(29,72,118,89)(30,65,119,90)(31,66,120,91)(32,67,113,92)(81,99,106,124)(82,100,107,125)(83,101,108,126)(84,102,109,127)(85,103,110,128)(86,104,111,121)(87,97,112,122)(88,98,105,123), (1,93,47,68)(2,96,48,71)(3,91,41,66)(4,94,42,69)(5,89,43,72)(6,92,44,67)(7,95,45,70)(8,90,46,65)(9,108,18,83)(10,111,19,86)(11,106,20,81)(12,109,21,84)(13,112,22,87)(14,107,23,82)(15,110,24,85)(16,105,17,88)(25,63,114,38)(26,58,115,33)(27,61,116,36)(28,64,117,39)(29,59,118,34)(30,62,119,37)(31,57,120,40)(32,60,113,35)(49,122,76,97)(50,125,77,100)(51,128,78,103)(52,123,79,98)(53,126,80,101)(54,121,73,104)(55,124,74,99)(56,127,75,102)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,104)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,76)(34,77)(35,78)(36,79)(37,80)(38,73)(39,74)(40,75)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(65,108)(66,109)(67,110)(68,111)(69,112)(70,105)(71,106)(72,107)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)(113,128)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127), (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), (1,63,47,38)(2,64,48,39)(3,57,41,40)(4,58,42,33)(5,59,43,34)(6,60,44,35)(7,61,45,36)(8,62,46,37)(9,80,18,53)(10,73,19,54)(11,74,20,55)(12,75,21,56)(13,76,22,49)(14,77,23,50)(15,78,24,51)(16,79,17,52)(25,68,114,93)(26,69,115,94)(27,70,116,95)(28,71,117,96)(29,72,118,89)(30,65,119,90)(31,66,120,91)(32,67,113,92)(81,99,106,124)(82,100,107,125)(83,101,108,126)(84,102,109,127)(85,103,110,128)(86,104,111,121)(87,97,112,122)(88,98,105,123), (1,93,47,68)(2,96,48,71)(3,91,41,66)(4,94,42,69)(5,89,43,72)(6,92,44,67)(7,95,45,70)(8,90,46,65)(9,108,18,83)(10,111,19,86)(11,106,20,81)(12,109,21,84)(13,112,22,87)(14,107,23,82)(15,110,24,85)(16,105,17,88)(25,63,114,38)(26,58,115,33)(27,61,116,36)(28,64,117,39)(29,59,118,34)(30,62,119,37)(31,57,120,40)(32,60,113,35)(49,122,76,97)(50,125,77,100)(51,128,78,103)(52,123,79,98)(53,126,80,101)(54,121,73,104)(55,124,74,99)(56,127,75,102) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,104),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,76),(34,77),(35,78),(36,79),(37,80),(38,73),(39,74),(40,75),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57),(65,108),(66,109),(67,110),(68,111),(69,112),(70,105),(71,106),(72,107),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95),(113,128),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(120,127)], [(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)], [(1,63,47,38),(2,64,48,39),(3,57,41,40),(4,58,42,33),(5,59,43,34),(6,60,44,35),(7,61,45,36),(8,62,46,37),(9,80,18,53),(10,73,19,54),(11,74,20,55),(12,75,21,56),(13,76,22,49),(14,77,23,50),(15,78,24,51),(16,79,17,52),(25,68,114,93),(26,69,115,94),(27,70,116,95),(28,71,117,96),(29,72,118,89),(30,65,119,90),(31,66,120,91),(32,67,113,92),(81,99,106,124),(82,100,107,125),(83,101,108,126),(84,102,109,127),(85,103,110,128),(86,104,111,121),(87,97,112,122),(88,98,105,123)], [(1,93,47,68),(2,96,48,71),(3,91,41,66),(4,94,42,69),(5,89,43,72),(6,92,44,67),(7,95,45,70),(8,90,46,65),(9,108,18,83),(10,111,19,86),(11,106,20,81),(12,109,21,84),(13,112,22,87),(14,107,23,82),(15,110,24,85),(16,105,17,88),(25,63,114,38),(26,58,115,33),(27,61,116,36),(28,64,117,39),(29,59,118,34),(30,62,119,37),(31,57,120,40),(32,60,113,35),(49,122,76,97),(50,125,77,100),(51,128,78,103),(52,123,79,98),(53,126,80,101),(54,121,73,104),(55,124,74,99),(56,127,75,102)])
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 | 12 | 5 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 0 | 11 | 13 |
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],[16,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,12,12,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,4,11,0,0,0,11,13] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | SD16 |
| kernel | C2×C8⋊3Q8 | C2×C4×C8 | C2×C4.Q8 | C8⋊3Q8 | C2×C4⋊Q8 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_3Q_8
% in TeX
G:=Group("C2xC8:3Q8"); // GroupNames label
G:=SmallGroup(128,1889);
// by ID
G=gap.SmallGroup(128,1889);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀