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, (C2×C4).81SD16, C4.22(C2×SD16), (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
Generators and relations for C2×C8⋊3Q8
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 >
Subgroups: 372 in 212 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C4×C8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C2×C4×C8, C2×C4.Q8, C8⋊3Q8, C2×C4⋊Q8, C2×C8⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C24, C4⋊Q8, C2×SD16, C22×D4, C22×Q8, C8⋊3Q8, C2×C4⋊Q8, C22×SD16, C2×C8⋊3Q8
►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 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 102)(26 103)(27 104)(28 97)(29 98)(30 99)(31 100)(32 101)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)
(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 63 47 40)(2 64 48 33)(3 57 41 34)(4 58 42 35)(5 59 43 36)(6 60 44 37)(7 61 45 38)(8 62 46 39)(9 78 24 53)(10 79 17 54)(11 80 18 55)(12 73 19 56)(13 74 20 49)(14 75 21 50)(15 76 22 51)(16 77 23 52)(25 66 120 93)(26 67 113 94)(27 68 114 95)(28 69 115 96)(29 70 116 89)(30 71 117 90)(31 72 118 91)(32 65 119 92)(81 99 108 126)(82 100 109 127)(83 101 110 128)(84 102 111 121)(85 103 112 122)(86 104 105 123)(87 97 106 124)(88 98 107 125)
(1 93 47 66)(2 96 48 69)(3 91 41 72)(4 94 42 67)(5 89 43 70)(6 92 44 65)(7 95 45 68)(8 90 46 71)(9 110 24 83)(10 105 17 86)(11 108 18 81)(12 111 19 84)(13 106 20 87)(14 109 21 82)(15 112 22 85)(16 107 23 88)(25 63 120 40)(26 58 113 35)(27 61 114 38)(28 64 115 33)(29 59 116 36)(30 62 117 39)(31 57 118 34)(32 60 119 37)(49 124 74 97)(50 127 75 100)(51 122 76 103)(52 125 77 98)(53 128 78 101)(54 123 79 104)(55 126 80 99)(56 121 73 102)
G:=sub<Sym(128)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,102)(26,103)(27,104)(28,97)(29,98)(30,99)(31,100)(32,101)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121), (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), (1,63,47,40)(2,64,48,33)(3,57,41,34)(4,58,42,35)(5,59,43,36)(6,60,44,37)(7,61,45,38)(8,62,46,39)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,66,120,93)(26,67,113,94)(27,68,114,95)(28,69,115,96)(29,70,116,89)(30,71,117,90)(31,72,118,91)(32,65,119,92)(81,99,108,126)(82,100,109,127)(83,101,110,128)(84,102,111,121)(85,103,112,122)(86,104,105,123)(87,97,106,124)(88,98,107,125), (1,93,47,66)(2,96,48,69)(3,91,41,72)(4,94,42,67)(5,89,43,70)(6,92,44,65)(7,95,45,68)(8,90,46,71)(9,110,24,83)(10,105,17,86)(11,108,18,81)(12,111,19,84)(13,106,20,87)(14,109,21,82)(15,112,22,85)(16,107,23,88)(25,63,120,40)(26,58,113,35)(27,61,114,38)(28,64,115,33)(29,59,116,36)(30,62,117,39)(31,57,118,34)(32,60,119,37)(49,124,74,97)(50,127,75,100)(51,122,76,103)(52,125,77,98)(53,128,78,101)(54,123,79,104)(55,126,80,99)(56,121,73,102)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,102)(26,103)(27,104)(28,97)(29,98)(30,99)(31,100)(32,101)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121), (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), (1,63,47,40)(2,64,48,33)(3,57,41,34)(4,58,42,35)(5,59,43,36)(6,60,44,37)(7,61,45,38)(8,62,46,39)(9,78,24,53)(10,79,17,54)(11,80,18,55)(12,73,19,56)(13,74,20,49)(14,75,21,50)(15,76,22,51)(16,77,23,52)(25,66,120,93)(26,67,113,94)(27,68,114,95)(28,69,115,96)(29,70,116,89)(30,71,117,90)(31,72,118,91)(32,65,119,92)(81,99,108,126)(82,100,109,127)(83,101,110,128)(84,102,111,121)(85,103,112,122)(86,104,105,123)(87,97,106,124)(88,98,107,125), (1,93,47,66)(2,96,48,69)(3,91,41,72)(4,94,42,67)(5,89,43,70)(6,92,44,65)(7,95,45,68)(8,90,46,71)(9,110,24,83)(10,105,17,86)(11,108,18,81)(12,111,19,84)(13,106,20,87)(14,109,21,82)(15,112,22,85)(16,107,23,88)(25,63,120,40)(26,58,113,35)(27,61,114,38)(28,64,115,33)(29,59,116,36)(30,62,117,39)(31,57,118,34)(32,60,119,37)(49,124,74,97)(50,127,75,100)(51,122,76,103)(52,125,77,98)(53,128,78,101)(54,123,79,104)(55,126,80,99)(56,121,73,102) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,102),(26,103),(27,104),(28,97),(29,98),(30,99),(31,100),(32,101),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(65,110),(66,111),(67,112),(68,105),(69,106),(70,107),(71,108),(72,109),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,121)], [(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)], [(1,63,47,40),(2,64,48,33),(3,57,41,34),(4,58,42,35),(5,59,43,36),(6,60,44,37),(7,61,45,38),(8,62,46,39),(9,78,24,53),(10,79,17,54),(11,80,18,55),(12,73,19,56),(13,74,20,49),(14,75,21,50),(15,76,22,51),(16,77,23,52),(25,66,120,93),(26,67,113,94),(27,68,114,95),(28,69,115,96),(29,70,116,89),(30,71,117,90),(31,72,118,91),(32,65,119,92),(81,99,108,126),(82,100,109,127),(83,101,110,128),(84,102,111,121),(85,103,112,122),(86,104,105,123),(87,97,106,124),(88,98,107,125)], [(1,93,47,66),(2,96,48,69),(3,91,41,72),(4,94,42,67),(5,89,43,70),(6,92,44,65),(7,95,45,68),(8,90,46,71),(9,110,24,83),(10,105,17,86),(11,108,18,81),(12,111,19,84),(13,106,20,87),(14,109,21,82),(15,112,22,85),(16,107,23,88),(25,63,120,40),(26,58,113,35),(27,61,114,38),(28,64,115,33),(29,59,116,36),(30,62,117,39),(31,57,118,34),(32,60,119,37),(49,124,74,97),(50,127,75,100),(51,122,76,103),(52,125,77,98),(53,128,78,101),(54,123,79,104),(55,126,80,99),(56,121,73,102)]])
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 |
Matrix representation of C2×C8⋊3Q8 ►in 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] >;
C2×C8⋊3Q8 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