direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊2Dic7, C14⋊3C4≀C2, (D4×C14)⋊8C4, (Q8×C14)⋊8C4, C4○D4⋊3Dic7, Q8⋊5(C2×Dic7), (C2×Q8)⋊6Dic7, (C2×D4)⋊8Dic7, D4⋊5(C2×Dic7), C4○D4.36D14, C28.452(C2×D4), (C2×C28).197D4, C28.85(C22×C4), C28.99(C22⋊C4), (C2×C28).481C23, (C4×Dic7)⋊63C22, (C22×C4).357D14, (C22×C14).113D4, C23.66(C7⋊D4), C4.Dic7⋊23C22, C4.33(C23.D7), C4.15(C22×Dic7), C22.5(C23.D7), (C22×C28).207C22, C7⋊4(C2×C4≀C2), (C7×C4○D4)⋊3C4, (C2×C4×Dic7)⋊4C2, (C7×D4)⋊18(C2×C4), (C7×Q8)⋊17(C2×C4), (C2×C4○D4).4D7, (C14×C4○D4).4C2, (C2×C14).39(C2×D4), C4.143(C2×C7⋊D4), (C2×C28).125(C2×C4), C14.84(C2×C22⋊C4), (C2×C4.Dic7)⋊21C2, (C2×C4).54(C2×Dic7), C22.11(C2×C7⋊D4), C2.20(C2×C23.D7), (C2×C4).282(C7⋊D4), (C7×C4○D4).41C22, (C2×C4).566(C22×D7), (C2×C14).117(C22⋊C4), SmallGroup(448,769)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊2Dic7
G = < a,b,c,d,e | a2=b4=d14=1, c2=b2, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 532 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C7, C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C14, C14 [×2], C14 [×4], C42 [×3], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic7 [×4], C28 [×4], C28 [×2], C2×C14 [×3], C2×C14 [×6], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C7⋊C8 [×2], C2×Dic7 [×6], C2×C28 [×6], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×C14, C22×C14, C2×C4≀C2, C2×C7⋊C8, C4.Dic7 [×2], C4.Dic7, C4×Dic7 [×2], C4×Dic7, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], Q8⋊2Dic7 [×4], C2×C4.Dic7, C2×C4×Dic7, C14×C4○D4, C2×Q8⋊2Dic7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic7 [×4], D14 [×3], C4≀C2 [×2], C2×C22⋊C4, C2×Dic7 [×6], C7⋊D4 [×4], C22×D7, C2×C4≀C2, C23.D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], Q8⋊2Dic7 [×2], C2×C23.D7, C2×Q8⋊2Dic7
►On 112 points
Generators in S112
(1 10)(2 11)(3 12)(4 13)(5 14)(6 8)(7 9)(15 39)(16 40)(17 41)(18 42)(19 36)(20 37)(21 38)(22 52)(23 53)(24 54)(25 55)(26 56)(27 50)(28 51)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(57 95)(58 96)(59 97)(60 98)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 99)
(1 22 49 15)(2 23 43 16)(3 24 44 17)(4 25 45 18)(5 26 46 19)(6 27 47 20)(7 28 48 21)(8 50 31 37)(9 51 32 38)(10 52 33 39)(11 53 34 40)(12 54 35 41)(13 55 29 42)(14 56 30 36)(57 80 64 73)(58 81 65 74)(59 82 66 75)(60 83 67 76)(61 84 68 77)(62 71 69 78)(63 72 70 79)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(1 99 49 106)(2 107 43 100)(3 101 44 108)(4 109 45 102)(5 103 46 110)(6 111 47 104)(7 105 48 112)(8 82 31 75)(9 76 32 83)(10 84 33 77)(11 78 34 71)(12 72 35 79)(13 80 29 73)(14 74 30 81)(15 92 22 85)(16 86 23 93)(17 94 24 87)(18 88 25 95)(19 96 26 89)(20 90 27 97)(21 98 28 91)(36 58 56 65)(37 66 50 59)(38 60 51 67)(39 68 52 61)(40 62 53 69)(41 70 54 63)(42 64 55 57)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 35)(7 34)(8 44)(9 43)(10 49)(11 48)(12 47)(13 46)(14 45)(15 52)(16 51)(17 50)(18 56)(19 55)(20 54)(21 53)(22 39)(23 38)(24 37)(25 36)(26 42)(27 41)(28 40)(57 103 64 110)(58 102 65 109)(59 101 66 108)(60 100 67 107)(61 99 68 106)(62 112 69 105)(63 111 70 104)(71 91 78 98)(72 90 79 97)(73 89 80 96)(74 88 81 95)(75 87 82 94)(76 86 83 93)(77 85 84 92)
G:=sub<Sym(112)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,8)(7,9)(15,39)(16,40)(17,41)(18,42)(19,36)(20,37)(21,38)(22,52)(23,53)(24,54)(25,55)(26,56)(27,50)(28,51)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(57,95)(58,96)(59,97)(60,98)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,99), (1,22,49,15)(2,23,43,16)(3,24,44,17)(4,25,45,18)(5,26,46,19)(6,27,47,20)(7,28,48,21)(8,50,31,37)(9,51,32,38)(10,52,33,39)(11,53,34,40)(12,54,35,41)(13,55,29,42)(14,56,30,36)(57,80,64,73)(58,81,65,74)(59,82,66,75)(60,83,67,76)(61,84,68,77)(62,71,69,78)(63,72,70,79)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (1,99,49,106)(2,107,43,100)(3,101,44,108)(4,109,45,102)(5,103,46,110)(6,111,47,104)(7,105,48,112)(8,82,31,75)(9,76,32,83)(10,84,33,77)(11,78,34,71)(12,72,35,79)(13,80,29,73)(14,74,30,81)(15,92,22,85)(16,86,23,93)(17,94,24,87)(18,88,25,95)(19,96,26,89)(20,90,27,97)(21,98,28,91)(36,58,56,65)(37,66,50,59)(38,60,51,67)(39,68,52,61)(40,62,53,69)(41,70,54,63)(42,64,55,57), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,44)(9,43)(10,49)(11,48)(12,47)(13,46)(14,45)(15,52)(16,51)(17,50)(18,56)(19,55)(20,54)(21,53)(22,39)(23,38)(24,37)(25,36)(26,42)(27,41)(28,40)(57,103,64,110)(58,102,65,109)(59,101,66,108)(60,100,67,107)(61,99,68,106)(62,112,69,105)(63,111,70,104)(71,91,78,98)(72,90,79,97)(73,89,80,96)(74,88,81,95)(75,87,82,94)(76,86,83,93)(77,85,84,92)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,8)(7,9)(15,39)(16,40)(17,41)(18,42)(19,36)(20,37)(21,38)(22,52)(23,53)(24,54)(25,55)(26,56)(27,50)(28,51)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(57,95)(58,96)(59,97)(60,98)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,99), (1,22,49,15)(2,23,43,16)(3,24,44,17)(4,25,45,18)(5,26,46,19)(6,27,47,20)(7,28,48,21)(8,50,31,37)(9,51,32,38)(10,52,33,39)(11,53,34,40)(12,54,35,41)(13,55,29,42)(14,56,30,36)(57,80,64,73)(58,81,65,74)(59,82,66,75)(60,83,67,76)(61,84,68,77)(62,71,69,78)(63,72,70,79)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (1,99,49,106)(2,107,43,100)(3,101,44,108)(4,109,45,102)(5,103,46,110)(6,111,47,104)(7,105,48,112)(8,82,31,75)(9,76,32,83)(10,84,33,77)(11,78,34,71)(12,72,35,79)(13,80,29,73)(14,74,30,81)(15,92,22,85)(16,86,23,93)(17,94,24,87)(18,88,25,95)(19,96,26,89)(20,90,27,97)(21,98,28,91)(36,58,56,65)(37,66,50,59)(38,60,51,67)(39,68,52,61)(40,62,53,69)(41,70,54,63)(42,64,55,57), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,44)(9,43)(10,49)(11,48)(12,47)(13,46)(14,45)(15,52)(16,51)(17,50)(18,56)(19,55)(20,54)(21,53)(22,39)(23,38)(24,37)(25,36)(26,42)(27,41)(28,40)(57,103,64,110)(58,102,65,109)(59,101,66,108)(60,100,67,107)(61,99,68,106)(62,112,69,105)(63,111,70,104)(71,91,78,98)(72,90,79,97)(73,89,80,96)(74,88,81,95)(75,87,82,94)(76,86,83,93)(77,85,84,92) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,13),(5,14),(6,8),(7,9),(15,39),(16,40),(17,41),(18,42),(19,36),(20,37),(21,38),(22,52),(23,53),(24,54),(25,55),(26,56),(27,50),(28,51),(29,45),(30,46),(31,47),(32,48),(33,49),(34,43),(35,44),(57,95),(58,96),(59,97),(60,98),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,100),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111),(83,112),(84,99)], [(1,22,49,15),(2,23,43,16),(3,24,44,17),(4,25,45,18),(5,26,46,19),(6,27,47,20),(7,28,48,21),(8,50,31,37),(9,51,32,38),(10,52,33,39),(11,53,34,40),(12,54,35,41),(13,55,29,42),(14,56,30,36),(57,80,64,73),(58,81,65,74),(59,82,66,75),(60,83,67,76),(61,84,68,77),(62,71,69,78),(63,72,70,79),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(1,99,49,106),(2,107,43,100),(3,101,44,108),(4,109,45,102),(5,103,46,110),(6,111,47,104),(7,105,48,112),(8,82,31,75),(9,76,32,83),(10,84,33,77),(11,78,34,71),(12,72,35,79),(13,80,29,73),(14,74,30,81),(15,92,22,85),(16,86,23,93),(17,94,24,87),(18,88,25,95),(19,96,26,89),(20,90,27,97),(21,98,28,91),(36,58,56,65),(37,66,50,59),(38,60,51,67),(39,68,52,61),(40,62,53,69),(41,70,54,63),(42,64,55,57)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,35),(7,34),(8,44),(9,43),(10,49),(11,48),(12,47),(13,46),(14,45),(15,52),(16,51),(17,50),(18,56),(19,55),(20,54),(21,53),(22,39),(23,38),(24,37),(25,36),(26,42),(27,41),(28,40),(57,103,64,110),(58,102,65,109),(59,101,66,108),(60,100,67,107),(61,99,68,106),(62,112,69,105),(63,111,70,104),(71,91,78,98),(72,90,79,97),(73,89,80,96),(74,88,81,95),(75,87,82,94),(76,86,83,93),(77,85,84,92)])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | ··· | 14 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D7 | D14 | Dic7 | Dic7 | Dic7 | D14 | C4≀C2 | C7⋊D4 | C7⋊D4 | Q8⋊2Dic7 |
| kernel | C2×Q8⋊2Dic7 | Q8⋊2Dic7 | C2×C4.Dic7 | C2×C4×Dic7 | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 8 | 18 | 6 | 12 |
Matrix representation of C2×Q8⋊2Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 98 | 0 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 20 | 105 | 0 | 0 |
| 36 | 93 | 0 | 0 |
| 0 | 0 | 79 | 108 |
| 0 | 0 | 5 | 34 |
| 1 | 0 | 0 | 0 |
| 5 | 112 | 0 | 0 |
| 0 | 0 | 9 | 112 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 40 | 98 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 104 | 1 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[98,38,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[20,36,0,0,105,93,0,0,0,0,79,5,0,0,108,34],[1,5,0,0,0,112,0,0,0,0,9,1,0,0,112,0],[1,40,0,0,0,98,0,0,0,0,112,104,0,0,0,1] >;
C2×Q8⋊2Dic7 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_2{\rm Dic}_7 % in TeX
G:=Group("C2xQ8:2Dic7"); // GroupNames label
G:=SmallGroup(448,769);
// by ID
G=gap.SmallGroup(448,769);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,136,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^14=1,c^2=b^2,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations