p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.5Q8, (C2×C4).28D8, (C2×C4).45SD16, C22.93(C2×D8), C2.8(C4.4D8), C23.945(C2×D4), (C22×C4).328D4, C42⋊9C4.20C2, C2.11(D4⋊Q8), C2.11(Q8⋊Q8), (C22×C8).90C22, C4.25(C42.C2), C4.22(C42⋊2C2), C22.4Q16.30C2, (C2×C42).396C22, C22.110(C2×SD16), (C22×C4).1479C23, C2.11(C22.D8), C22.97(C4.4D4), C22.122(C22⋊Q8), C2.11(C23.47D4), C22.150(C8.C22), C22.7C42.15C2, C23.65C23.26C2, C2.8(C42.30C22), C2.5(C23.83C23), C22.132(C22.D4), (C2×C4).296(C2×Q8), (C2×C4).785(C4○D4), (C2×C4⋊C4).168C22, SmallGroup(128,831)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).28D8
G = < a,b,c,d | a2=b4=c8=1, d2=ab2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 224 in 107 conjugacy classes, 50 normal (28 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊9C4, C23.65C23, (C2×C4).28D8
Quotients: C1, C2, C22, D4, Q8, C23, D8, SD16, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C8.C22, C23.83C23, D4⋊Q8, Q8⋊Q8, C22.D8, C23.47D4, C4.4D8, C42.30C22, (C2×C4).28D8
►Regular action on 128 points
Generators in S128
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)(89 126)(90 127)(91 128)(92 121)(93 122)(94 123)(95 124)(96 125)(105 114)(106 115)(107 116)(108 117)(109 118)(110 119)(111 120)(112 113)
(1 94 87 98)(2 51 88 124)(3 96 81 100)(4 53 82 126)(5 90 83 102)(6 55 84 128)(7 92 85 104)(8 49 86 122)(9 76 41 112)(10 114 42 70)(11 78 43 106)(12 116 44 72)(13 80 45 108)(14 118 46 66)(15 74 47 110)(16 120 48 68)(17 79 39 107)(18 117 40 65)(19 73 33 109)(20 119 34 67)(21 75 35 111)(22 113 36 69)(23 77 37 105)(24 115 38 71)(25 127 59 54)(26 103 60 91)(27 121 61 56)(28 97 62 93)(29 123 63 50)(30 99 64 95)(31 125 57 52)(32 101 58 89)
(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 33 63 46)(2 13 64 18)(3 39 57 44)(4 11 58 24)(5 37 59 42)(6 9 60 22)(7 35 61 48)(8 15 62 20)(10 83 23 25)(12 81 17 31)(14 87 19 29)(16 85 21 27)(26 36 84 41)(28 34 86 47)(30 40 88 45)(32 38 82 43)(49 110 93 67)(50 118 94 73)(51 108 95 65)(52 116 96 79)(53 106 89 71)(54 114 90 77)(55 112 91 69)(56 120 92 75)(66 98 109 123)(68 104 111 121)(70 102 105 127)(72 100 107 125)(74 97 119 122)(76 103 113 128)(78 101 115 126)(80 99 117 124)
G:=sub<Sym(128)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79)(89,126)(90,127)(91,128)(92,121)(93,122)(94,123)(95,124)(96,125)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,113), (1,94,87,98)(2,51,88,124)(3,96,81,100)(4,53,82,126)(5,90,83,102)(6,55,84,128)(7,92,85,104)(8,49,86,122)(9,76,41,112)(10,114,42,70)(11,78,43,106)(12,116,44,72)(13,80,45,108)(14,118,46,66)(15,74,47,110)(16,120,48,68)(17,79,39,107)(18,117,40,65)(19,73,33,109)(20,119,34,67)(21,75,35,111)(22,113,36,69)(23,77,37,105)(24,115,38,71)(25,127,59,54)(26,103,60,91)(27,121,61,56)(28,97,62,93)(29,123,63,50)(30,99,64,95)(31,125,57,52)(32,101,58,89), (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), (1,33,63,46)(2,13,64,18)(3,39,57,44)(4,11,58,24)(5,37,59,42)(6,9,60,22)(7,35,61,48)(8,15,62,20)(10,83,23,25)(12,81,17,31)(14,87,19,29)(16,85,21,27)(26,36,84,41)(28,34,86,47)(30,40,88,45)(32,38,82,43)(49,110,93,67)(50,118,94,73)(51,108,95,65)(52,116,96,79)(53,106,89,71)(54,114,90,77)(55,112,91,69)(56,120,92,75)(66,98,109,123)(68,104,111,121)(70,102,105,127)(72,100,107,125)(74,97,119,122)(76,103,113,128)(78,101,115,126)(80,99,117,124)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79)(89,126)(90,127)(91,128)(92,121)(93,122)(94,123)(95,124)(96,125)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,113), (1,94,87,98)(2,51,88,124)(3,96,81,100)(4,53,82,126)(5,90,83,102)(6,55,84,128)(7,92,85,104)(8,49,86,122)(9,76,41,112)(10,114,42,70)(11,78,43,106)(12,116,44,72)(13,80,45,108)(14,118,46,66)(15,74,47,110)(16,120,48,68)(17,79,39,107)(18,117,40,65)(19,73,33,109)(20,119,34,67)(21,75,35,111)(22,113,36,69)(23,77,37,105)(24,115,38,71)(25,127,59,54)(26,103,60,91)(27,121,61,56)(28,97,62,93)(29,123,63,50)(30,99,64,95)(31,125,57,52)(32,101,58,89), (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), (1,33,63,46)(2,13,64,18)(3,39,57,44)(4,11,58,24)(5,37,59,42)(6,9,60,22)(7,35,61,48)(8,15,62,20)(10,83,23,25)(12,81,17,31)(14,87,19,29)(16,85,21,27)(26,36,84,41)(28,34,86,47)(30,40,88,45)(32,38,82,43)(49,110,93,67)(50,118,94,73)(51,108,95,65)(52,116,96,79)(53,106,89,71)(54,114,90,77)(55,112,91,69)(56,120,92,75)(66,98,109,123)(68,104,111,121)(70,102,105,127)(72,100,107,125)(74,97,119,122)(76,103,113,128)(78,101,115,126)(80,99,117,124) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,80),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79),(89,126),(90,127),(91,128),(92,121),(93,122),(94,123),(95,124),(96,125),(105,114),(106,115),(107,116),(108,117),(109,118),(110,119),(111,120),(112,113)], [(1,94,87,98),(2,51,88,124),(3,96,81,100),(4,53,82,126),(5,90,83,102),(6,55,84,128),(7,92,85,104),(8,49,86,122),(9,76,41,112),(10,114,42,70),(11,78,43,106),(12,116,44,72),(13,80,45,108),(14,118,46,66),(15,74,47,110),(16,120,48,68),(17,79,39,107),(18,117,40,65),(19,73,33,109),(20,119,34,67),(21,75,35,111),(22,113,36,69),(23,77,37,105),(24,115,38,71),(25,127,59,54),(26,103,60,91),(27,121,61,56),(28,97,62,93),(29,123,63,50),(30,99,64,95),(31,125,57,52),(32,101,58,89)], [(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)], [(1,33,63,46),(2,13,64,18),(3,39,57,44),(4,11,58,24),(5,37,59,42),(6,9,60,22),(7,35,61,48),(8,15,62,20),(10,83,23,25),(12,81,17,31),(14,87,19,29),(16,85,21,27),(26,36,84,41),(28,34,86,47),(30,40,88,45),(32,38,82,43),(49,110,93,67),(50,118,94,73),(51,108,95,65),(52,116,96,79),(53,106,89,71),(54,114,90,77),(55,112,91,69),(56,120,92,75),(66,98,109,123),(68,104,111,121),(70,102,105,127),(72,100,107,125),(74,97,119,122),(76,103,113,128),(78,101,115,126),(80,99,117,124)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D8 | SD16 | C4○D4 | C8.C22 |
| kernel | (C2×C4).28D8 | C22.7C42 | C22.4Q16 | C42⋊9C4 | C23.65C23 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 2 |
Matrix representation of (C2×C4).28D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 5 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,5,0,0,0,0,12,14,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[14,5,0,0,0,0,12,3,0,0,0,0,0,0,16,0,0,0,0,0,8,1,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,4,0,0,0,0,0,2,13,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
(C2×C4).28D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{28}D_8 % in TeX
G:=Group("(C2xC4).28D8"); // GroupNames label
G:=SmallGroup(128,831);
// by ID
G=gap.SmallGroup(128,831);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=a*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations