p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).24Q8, C4⋊C4.112D4, C2.11(C8⋊Q8), C23.935(C2×D4), (C22×C4).163D4, C22.53(C4⋊Q8), C4.13(C22⋊Q8), C2.6(C8.5Q8), C42⋊8C4.16C2, C4.21(C42.C2), C2.23(D4.2D4), C22.125(C4○D8), C22.4Q16.41C2, (C22×C8).119C22, (C2×C42).386C22, C2.23(Q8.D4), C22.258(C4⋊D4), C22.155(C8⋊C22), (C22×C4).1469C23, C2.13(C23.19D4), C2.13(C23.20D4), C22.144(C8.C22), C22.7C42.28C2, C23.65C23.22C2, C2.7(C23.81C23), C22.125(C22.D4), (C2×C4).222(C2×Q8), (C2×C2.D8).16C2, (C2×C4.Q8).26C2, (C2×C4).1065(C2×D4), (C2×C4).888(C4○D4), (C2×C4⋊C4).156C22, SmallGroup(128,817)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).24Q8
G = < a,b,c,d | a2=b8=c4=1, d2=ab4c2, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=b3, dcd-1=ab4c-1 >
Subgroups: 216 in 107 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C42⋊8C4, C23.65C23, C2×C4.Q8, C2×C2.D8, (C2×C8).24Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C4○D8, C8⋊C22, C8.C22, C23.81C23, D4.2D4, Q8.D4, C23.19D4, C23.20D4, C8.5Q8, C8⋊Q8, (C2×C8).24Q8
►Regular action on 128 points
Generators in S128
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(41 104)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 109)(66 110)(67 111)(68 112)(69 105)(70 106)(71 107)(72 108)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 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 58 12 106)(2 117 13 65)(3 64 14 112)(4 115 15 71)(5 62 16 110)(6 113 9 69)(7 60 10 108)(8 119 11 67)(17 88 47 29)(18 92 48 37)(19 86 41 27)(20 90 42 35)(21 84 43 25)(22 96 44 33)(23 82 45 31)(24 94 46 39)(26 78 85 101)(28 76 87 99)(30 74 81 97)(32 80 83 103)(34 79 89 102)(36 77 91 100)(38 75 93 98)(40 73 95 104)(49 111 126 63)(50 70 127 114)(51 109 128 61)(52 68 121 120)(53 107 122 59)(54 66 123 118)(55 105 124 57)(56 72 125 116)
(1 97 54 24)(2 100 55 19)(3 103 56 22)(4 98 49 17)(5 101 50 20)(6 104 51 23)(7 99 52 18)(8 102 53 21)(9 73 128 45)(10 76 121 48)(11 79 122 43)(12 74 123 46)(13 77 124 41)(14 80 125 44)(15 75 126 47)(16 78 127 42)(25 59 89 67)(26 62 90 70)(27 57 91 65)(28 60 92 68)(29 63 93 71)(30 58 94 66)(31 61 95 69)(32 64 96 72)(33 116 83 112)(34 119 84 107)(35 114 85 110)(36 117 86 105)(37 120 87 108)(38 115 88 111)(39 118 81 106)(40 113 82 109)
G:=sub<Sym(128)| (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,109)(66,110)(67,111)(68,112)(69,105)(70,106)(71,107)(72,108)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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), (1,58,12,106)(2,117,13,65)(3,64,14,112)(4,115,15,71)(5,62,16,110)(6,113,9,69)(7,60,10,108)(8,119,11,67)(17,88,47,29)(18,92,48,37)(19,86,41,27)(20,90,42,35)(21,84,43,25)(22,96,44,33)(23,82,45,31)(24,94,46,39)(26,78,85,101)(28,76,87,99)(30,74,81,97)(32,80,83,103)(34,79,89,102)(36,77,91,100)(38,75,93,98)(40,73,95,104)(49,111,126,63)(50,70,127,114)(51,109,128,61)(52,68,121,120)(53,107,122,59)(54,66,123,118)(55,105,124,57)(56,72,125,116), (1,97,54,24)(2,100,55,19)(3,103,56,22)(4,98,49,17)(5,101,50,20)(6,104,51,23)(7,99,52,18)(8,102,53,21)(9,73,128,45)(10,76,121,48)(11,79,122,43)(12,74,123,46)(13,77,124,41)(14,80,125,44)(15,75,126,47)(16,78,127,42)(25,59,89,67)(26,62,90,70)(27,57,91,65)(28,60,92,68)(29,63,93,71)(30,58,94,66)(31,61,95,69)(32,64,96,72)(33,116,83,112)(34,119,84,107)(35,114,85,110)(36,117,86,105)(37,120,87,108)(38,115,88,111)(39,118,81,106)(40,113,82,109)>;
G:=Group( (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,109)(66,110)(67,111)(68,112)(69,105)(70,106)(71,107)(72,108)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (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), (1,58,12,106)(2,117,13,65)(3,64,14,112)(4,115,15,71)(5,62,16,110)(6,113,9,69)(7,60,10,108)(8,119,11,67)(17,88,47,29)(18,92,48,37)(19,86,41,27)(20,90,42,35)(21,84,43,25)(22,96,44,33)(23,82,45,31)(24,94,46,39)(26,78,85,101)(28,76,87,99)(30,74,81,97)(32,80,83,103)(34,79,89,102)(36,77,91,100)(38,75,93,98)(40,73,95,104)(49,111,126,63)(50,70,127,114)(51,109,128,61)(52,68,121,120)(53,107,122,59)(54,66,123,118)(55,105,124,57)(56,72,125,116), (1,97,54,24)(2,100,55,19)(3,103,56,22)(4,98,49,17)(5,101,50,20)(6,104,51,23)(7,99,52,18)(8,102,53,21)(9,73,128,45)(10,76,121,48)(11,79,122,43)(12,74,123,46)(13,77,124,41)(14,80,125,44)(15,75,126,47)(16,78,127,42)(25,59,89,67)(26,62,90,70)(27,57,91,65)(28,60,92,68)(29,63,93,71)(30,58,94,66)(31,61,95,69)(32,64,96,72)(33,116,83,112)(34,119,84,107)(35,114,85,110)(36,117,86,105)(37,120,87,108)(38,115,88,111)(39,118,81,106)(40,113,82,109) );
G=PermutationGroup([[(1,127),(2,128),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,79),(18,80),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(41,104),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,109),(66,110),(67,111),(68,112),(69,105),(70,106),(71,107),(72,108),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89)], [(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)], [(1,58,12,106),(2,117,13,65),(3,64,14,112),(4,115,15,71),(5,62,16,110),(6,113,9,69),(7,60,10,108),(8,119,11,67),(17,88,47,29),(18,92,48,37),(19,86,41,27),(20,90,42,35),(21,84,43,25),(22,96,44,33),(23,82,45,31),(24,94,46,39),(26,78,85,101),(28,76,87,99),(30,74,81,97),(32,80,83,103),(34,79,89,102),(36,77,91,100),(38,75,93,98),(40,73,95,104),(49,111,126,63),(50,70,127,114),(51,109,128,61),(52,68,121,120),(53,107,122,59),(54,66,123,118),(55,105,124,57),(56,72,125,116)], [(1,97,54,24),(2,100,55,19),(3,103,56,22),(4,98,49,17),(5,101,50,20),(6,104,51,23),(7,99,52,18),(8,102,53,21),(9,73,128,45),(10,76,121,48),(11,79,122,43),(12,74,123,46),(13,77,124,41),(14,80,125,44),(15,75,126,47),(16,78,127,42),(25,59,89,67),(26,62,90,70),(27,57,91,65),(28,60,92,68),(29,63,93,71),(30,58,94,66),(31,61,95,69),(32,64,96,72),(33,116,83,112),(34,119,84,107),(35,114,85,110),(36,117,86,105),(37,120,87,108),(38,115,88,111),(39,118,81,106),(40,113,82,109)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | (C2×C8).24Q8 | C22.7C42 | C22.4Q16 | C42⋊8C4 | C23.65C23 | C2×C4.Q8 | C2×C2.D8 | C4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 6 | 8 | 1 | 1 |
Matrix representation of (C2×C8).24Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,11,0,0,0,0,11,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,8,1,0,0,0,0,0,0,10,1,0,0,0,0,1,7] >;
(C2×C8).24Q8 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{24}Q_8 % in TeX
G:=Group("(C2xC8).24Q8"); // GroupNames label
G:=SmallGroup(128,817);
// by ID
G=gap.SmallGroup(128,817);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a*b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=b^3,d*c*d^-1=a*b^4*c^-1>;
// generators/relations