direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C22⋊D8, D4⋊2(C7×D4), (C2×C14)⋊7D8, (C2×D8)⋊1C14, (C7×D4)⋊20D4, C2.4(C14×D8), C22⋊2(C7×D8), (C14×D8)⋊15C2, C4⋊D4⋊1C14, C22⋊C8⋊3C14, C4.21(D4×C14), C14.76(C2×D8), D4⋊C4⋊4C14, (C2×C56)⋊28C22, (C2×C28).317D4, C28.382(C2×D4), (C22×D4)⋊2C14, C23.41(C7×D4), C14.94C22≀C2, (D4×C14)⋊27C22, C22.77(D4×C14), (C2×C28).912C23, (C22×C14).163D4, C14.131(C8⋊C22), (C22×C28).419C22, C4⋊C4⋊1(C2×C14), (C2×C8)⋊1(C2×C14), (D4×C2×C14)⋊14C2, (C2×D4)⋊1(C2×C14), (C2×C4).26(C7×D4), C2.6(C7×C8⋊C22), (C7×C4⋊D4)⋊28C2, (C7×C4⋊C4)⋊35C22, (C7×C22⋊C8)⋊20C2, C2.8(C7×C22≀C2), (C7×D4⋊C4)⋊27C2, (C2×C14).633(C2×D4), (C2×C4).87(C22×C14), (C22×C4).37(C2×C14), SmallGroup(448,855)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22⋊D8
G = < a,b,c,d,e | a7=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 450 in 198 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C22⋊D8, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C22×C28, D4×C14, D4×C14, D4×C14, C23×C14, C7×C22⋊C8, C7×D4⋊C4, C7×C4⋊D4, C14×D8, D4×C2×C14, C7×C22⋊D8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C22≀C2, C2×D8, C8⋊C22, C7×D4, C22×C14, C22⋊D8, C7×D8, D4×C14, C7×C22≀C2, C14×D8, C7×C8⋊C22, C7×C22⋊D8
►On 112 points
Generators in S112
(1 67 9 85 28 77 20)(2 68 10 86 29 78 21)(3 69 11 87 30 79 22)(4 70 12 88 31 80 23)(5 71 13 81 32 73 24)(6 72 14 82 25 74 17)(7 65 15 83 26 75 18)(8 66 16 84 27 76 19)(33 64 106 49 98 41 90)(34 57 107 50 99 42 91)(35 58 108 51 100 43 92)(36 59 109 52 101 44 93)(37 60 110 53 102 45 94)(38 61 111 54 103 46 95)(39 62 112 55 104 47 96)(40 63 105 56 97 48 89)
(1 5)(2 35)(3 7)(4 37)(6 39)(8 33)(9 13)(10 108)(11 15)(12 110)(14 112)(16 106)(17 96)(18 22)(19 90)(20 24)(21 92)(23 94)(25 104)(26 30)(27 98)(28 32)(29 100)(31 102)(34 38)(36 40)(41 76)(42 46)(43 78)(44 48)(45 80)(47 74)(49 84)(50 54)(51 86)(52 56)(53 88)(55 82)(57 61)(58 68)(59 63)(60 70)(62 72)(64 66)(65 69)(67 71)(73 77)(75 79)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(1 38)(2 39)(3 40)(4 33)(5 34)(6 35)(7 36)(8 37)(9 111)(10 112)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 92)(18 93)(19 94)(20 95)(21 96)(22 89)(23 90)(24 91)(25 100)(26 101)(27 102)(28 103)(29 104)(30 97)(31 98)(32 99)(41 80)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 88)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 24)(25 30)(26 29)(27 28)(31 32)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 64)(58 63)(59 62)(60 61)(65 68)(66 67)(69 72)(70 71)(73 80)(74 79)(75 78)(76 77)(81 88)(82 87)(83 86)(84 85)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)
G:=sub<Sym(112)| (1,67,9,85,28,77,20)(2,68,10,86,29,78,21)(3,69,11,87,30,79,22)(4,70,12,88,31,80,23)(5,71,13,81,32,73,24)(6,72,14,82,25,74,17)(7,65,15,83,26,75,18)(8,66,16,84,27,76,19)(33,64,106,49,98,41,90)(34,57,107,50,99,42,91)(35,58,108,51,100,43,92)(36,59,109,52,101,44,93)(37,60,110,53,102,45,94)(38,61,111,54,103,46,95)(39,62,112,55,104,47,96)(40,63,105,56,97,48,89), (1,5)(2,35)(3,7)(4,37)(6,39)(8,33)(9,13)(10,108)(11,15)(12,110)(14,112)(16,106)(17,96)(18,22)(19,90)(20,24)(21,92)(23,94)(25,104)(26,30)(27,98)(28,32)(29,100)(31,102)(34,38)(36,40)(41,76)(42,46)(43,78)(44,48)(45,80)(47,74)(49,84)(50,54)(51,86)(52,56)(53,88)(55,82)(57,61)(58,68)(59,63)(60,70)(62,72)(64,66)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,111)(10,112)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,100)(26,101)(27,102)(28,103)(29,104)(30,97)(31,98)(32,99)(41,80)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,88)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,24)(25,30)(26,29)(27,28)(31,32)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,64)(58,63)(59,62)(60,61)(65,68)(66,67)(69,72)(70,71)(73,80)(74,79)(75,78)(76,77)(81,88)(82,87)(83,86)(84,85)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)>;
G:=Group( (1,67,9,85,28,77,20)(2,68,10,86,29,78,21)(3,69,11,87,30,79,22)(4,70,12,88,31,80,23)(5,71,13,81,32,73,24)(6,72,14,82,25,74,17)(7,65,15,83,26,75,18)(8,66,16,84,27,76,19)(33,64,106,49,98,41,90)(34,57,107,50,99,42,91)(35,58,108,51,100,43,92)(36,59,109,52,101,44,93)(37,60,110,53,102,45,94)(38,61,111,54,103,46,95)(39,62,112,55,104,47,96)(40,63,105,56,97,48,89), (1,5)(2,35)(3,7)(4,37)(6,39)(8,33)(9,13)(10,108)(11,15)(12,110)(14,112)(16,106)(17,96)(18,22)(19,90)(20,24)(21,92)(23,94)(25,104)(26,30)(27,98)(28,32)(29,100)(31,102)(34,38)(36,40)(41,76)(42,46)(43,78)(44,48)(45,80)(47,74)(49,84)(50,54)(51,86)(52,56)(53,88)(55,82)(57,61)(58,68)(59,63)(60,70)(62,72)(64,66)(65,69)(67,71)(73,77)(75,79)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,111)(10,112)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,100)(26,101)(27,102)(28,103)(29,104)(30,97)(31,98)(32,99)(41,80)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,88)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,24)(25,30)(26,29)(27,28)(31,32)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,64)(58,63)(59,62)(60,61)(65,68)(66,67)(69,72)(70,71)(73,80)(74,79)(75,78)(76,77)(81,88)(82,87)(83,86)(84,85)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111) );
G=PermutationGroup([[(1,67,9,85,28,77,20),(2,68,10,86,29,78,21),(3,69,11,87,30,79,22),(4,70,12,88,31,80,23),(5,71,13,81,32,73,24),(6,72,14,82,25,74,17),(7,65,15,83,26,75,18),(8,66,16,84,27,76,19),(33,64,106,49,98,41,90),(34,57,107,50,99,42,91),(35,58,108,51,100,43,92),(36,59,109,52,101,44,93),(37,60,110,53,102,45,94),(38,61,111,54,103,46,95),(39,62,112,55,104,47,96),(40,63,105,56,97,48,89)], [(1,5),(2,35),(3,7),(4,37),(6,39),(8,33),(9,13),(10,108),(11,15),(12,110),(14,112),(16,106),(17,96),(18,22),(19,90),(20,24),(21,92),(23,94),(25,104),(26,30),(27,98),(28,32),(29,100),(31,102),(34,38),(36,40),(41,76),(42,46),(43,78),(44,48),(45,80),(47,74),(49,84),(50,54),(51,86),(52,56),(53,88),(55,82),(57,61),(58,68),(59,63),(60,70),(62,72),(64,66),(65,69),(67,71),(73,77),(75,79),(81,85),(83,87),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)], [(1,38),(2,39),(3,40),(4,33),(5,34),(6,35),(7,36),(8,37),(9,111),(10,112),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,92),(18,93),(19,94),(20,95),(21,96),(22,89),(23,90),(24,91),(25,100),(26,101),(27,102),(28,103),(29,104),(30,97),(31,98),(32,99),(41,80),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,88),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,22),(18,21),(19,20),(23,24),(25,30),(26,29),(27,28),(31,32),(33,34),(35,40),(36,39),(37,38),(41,42),(43,48),(44,47),(45,46),(49,50),(51,56),(52,55),(53,54),(57,64),(58,63),(59,62),(60,61),(65,68),(66,67),(69,72),(70,71),(73,80),(74,79),(75,78),(76,77),(81,88),(82,87),(83,86),(84,85),(89,92),(90,91),(93,96),(94,95),(97,100),(98,99),(101,104),(102,103),(105,108),(106,107),(109,112),(110,111)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BB | 14BC | ··· | 14BH | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | D8 | C7×D4 | C7×D4 | C7×D4 | C7×D8 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C22⋊D8 | C7×C22⋊C8 | C7×D4⋊C4 | C7×C4⋊D4 | C14×D8 | D4×C2×C14 | C22⋊D8 | C22⋊C8 | D4⋊C4 | C4⋊D4 | C2×D8 | C22×D4 | C2×C28 | C7×D4 | C22×C14 | C2×C14 | C2×C4 | D4 | C23 | C22 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 12 | 6 | 1 | 4 | 1 | 4 | 6 | 24 | 6 | 24 | 1 | 6 |
Matrix representation of C7×C22⋊D8 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 82 | 31 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[82,82,0,0,31,82,0,0,0,0,0,112,0,0,1,0],[82,31,0,0,31,31,0,0,0,0,0,1,0,0,1,0] >;
C7×C22⋊D8 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes D_8
% in TeX
G:=Group("C7xC2^2:D8"); // GroupNames label
G:=SmallGroup(448,855);
// by ID
G=gap.SmallGroup(448,855);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations