direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C22⋊SD16, D4.6(C7×D4), C22⋊C8⋊9C14, (C7×D4).40D4, C4.24(D4×C14), C22⋊Q8⋊1C14, D4⋊C4⋊9C14, (C2×C56)⋊34C22, (C2×SD16)⋊9C14, (C2×C14)⋊10SD16, C28.385(C2×D4), (C2×C28).319D4, C2.6(C14×SD16), C22⋊2(C7×SD16), C23.43(C7×D4), C14.97C22≀C2, (C14×SD16)⋊26C2, C14.86(C2×SD16), (Q8×C14)⋊26C22, (C22×D4).8C14, C22.80(D4×C14), (C2×C28).915C23, (C22×C14).165D4, C14.133(C8⋊C22), (D4×C14).295C22, (C22×C28).422C22, C4⋊C4⋊2(C2×C14), (C2×C8)⋊6(C2×C14), (C2×Q8)⋊1(C2×C14), (D4×C2×C14).20C2, (C2×C4).28(C7×D4), C2.8(C7×C8⋊C22), (C7×C4⋊C4)⋊36C22, (C7×C22⋊C8)⋊26C2, (C7×C22⋊Q8)⋊28C2, (C7×D4⋊C4)⋊33C2, C2.11(C7×C22≀C2), (C2×D4).53(C2×C14), (C2×C14).636(C2×D4), (C22×C4).40(C2×C14), (C2×C4).90(C22×C14), SmallGroup(448,858)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22⋊SD16
G = < a,b,c,d,e | a7=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 402 in 188 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, C22×C14, C22⋊SD16, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C22×C28, D4×C14, D4×C14, Q8×C14, C23×C14, C7×C22⋊C8, C7×D4⋊C4, C7×C22⋊Q8, C14×SD16, D4×C2×C14, C7×C22⋊SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C2×C14, C22≀C2, C2×SD16, C8⋊C22, C7×D4, C22×C14, C22⋊SD16, C7×SD16, D4×C14, C7×C22≀C2, C14×SD16, C7×C8⋊C22, C7×C22⋊SD16
►On 112 points
Generators in S112
(1 10 83 26 75 18 67)(2 11 84 27 76 19 68)(3 12 85 28 77 20 69)(4 13 86 29 78 21 70)(5 14 87 30 79 22 71)(6 15 88 31 80 23 72)(7 16 81 32 73 24 65)(8 9 82 25 74 17 66)(33 63 106 49 98 41 90)(34 64 107 50 99 42 91)(35 57 108 51 100 43 92)(36 58 109 52 101 44 93)(37 59 110 53 102 45 94)(38 60 111 54 103 46 95)(39 61 112 55 104 47 96)(40 62 105 56 97 48 89)
(1 5)(2 39)(3 7)(4 33)(6 35)(8 37)(9 59)(10 14)(11 61)(12 16)(13 63)(15 57)(17 45)(18 22)(19 47)(20 24)(21 41)(23 43)(25 53)(26 30)(27 55)(28 32)(29 49)(31 51)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 69)(66 94)(67 71)(68 96)(70 90)(72 92)(73 77)(74 102)(75 79)(76 104)(78 98)(80 100)(81 85)(82 110)(83 87)(84 112)(86 106)(88 108)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 59)(58 62)(61 63)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)(81 85)(82 88)(84 86)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)
G:=sub<Sym(112)| (1,10,83,26,75,18,67)(2,11,84,27,76,19,68)(3,12,85,28,77,20,69)(4,13,86,29,78,21,70)(5,14,87,30,79,22,71)(6,15,88,31,80,23,72)(7,16,81,32,73,24,65)(8,9,82,25,74,17,66)(33,63,106,49,98,41,90)(34,64,107,50,99,42,91)(35,57,108,51,100,43,92)(36,58,109,52,101,44,93)(37,59,110,53,102,45,94)(38,60,111,54,103,46,95)(39,61,112,55,104,47,96)(40,62,105,56,97,48,89), (1,5)(2,39)(3,7)(4,33)(6,35)(8,37)(9,59)(10,14)(11,61)(12,16)(13,63)(15,57)(17,45)(18,22)(19,47)(20,24)(21,41)(23,43)(25,53)(26,30)(27,55)(28,32)(29,49)(31,51)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,69)(66,94)(67,71)(68,96)(70,90)(72,92)(73,77)(74,102)(75,79)(76,104)(78,98)(80,100)(81,85)(82,110)(83,87)(84,112)(86,106)(88,108)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)>;
G:=Group( (1,10,83,26,75,18,67)(2,11,84,27,76,19,68)(3,12,85,28,77,20,69)(4,13,86,29,78,21,70)(5,14,87,30,79,22,71)(6,15,88,31,80,23,72)(7,16,81,32,73,24,65)(8,9,82,25,74,17,66)(33,63,106,49,98,41,90)(34,64,107,50,99,42,91)(35,57,108,51,100,43,92)(36,58,109,52,101,44,93)(37,59,110,53,102,45,94)(38,60,111,54,103,46,95)(39,61,112,55,104,47,96)(40,62,105,56,97,48,89), (1,5)(2,39)(3,7)(4,33)(6,35)(8,37)(9,59)(10,14)(11,61)(12,16)(13,63)(15,57)(17,45)(18,22)(19,47)(20,24)(21,41)(23,43)(25,53)(26,30)(27,55)(28,32)(29,49)(31,51)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,69)(66,94)(67,71)(68,96)(70,90)(72,92)(73,77)(74,102)(75,79)(76,104)(78,98)(80,100)(81,85)(82,110)(83,87)(84,112)(86,106)(88,108)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110) );
G=PermutationGroup([[(1,10,83,26,75,18,67),(2,11,84,27,76,19,68),(3,12,85,28,77,20,69),(4,13,86,29,78,21,70),(5,14,87,30,79,22,71),(6,15,88,31,80,23,72),(7,16,81,32,73,24,65),(8,9,82,25,74,17,66),(33,63,106,49,98,41,90),(34,64,107,50,99,42,91),(35,57,108,51,100,43,92),(36,58,109,52,101,44,93),(37,59,110,53,102,45,94),(38,60,111,54,103,46,95),(39,61,112,55,104,47,96),(40,62,105,56,97,48,89)], [(1,5),(2,39),(3,7),(4,33),(6,35),(8,37),(9,59),(10,14),(11,61),(12,16),(13,63),(15,57),(17,45),(18,22),(19,47),(20,24),(21,41),(23,43),(25,53),(26,30),(27,55),(28,32),(29,49),(31,51),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(65,69),(66,94),(67,71),(68,96),(70,90),(72,92),(73,77),(74,102),(75,79),(76,104),(78,98),(80,100),(81,85),(82,110),(83,87),(84,112),(86,106),(88,108),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(57,59),(58,62),(61,63),(65,69),(66,72),(68,70),(73,77),(74,80),(76,78),(81,85),(82,88),(84,86),(89,93),(90,96),(92,94),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BB | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 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 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | SD16 | C7×D4 | C7×D4 | C7×D4 | C7×SD16 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C22⋊SD16 | C7×C22⋊C8 | C7×D4⋊C4 | C7×C22⋊Q8 | C14×SD16 | D4×C2×C14 | C22⋊SD16 | C22⋊C8 | D4⋊C4 | C22⋊Q8 | C2×SD16 | 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⋊SD16 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 56 | 0 | 0 | 0 |
| 0 | 0 | 13 | 100 |
| 0 | 0 | 13 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[0,56,0,0,2,0,0,0,0,0,13,13,0,0,100,13],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112] >;
C7×C22⋊SD16 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes {\rm SD}_{16} % in TeX
G:=Group("C7xC2^2:SD16"); // GroupNames label
G:=SmallGroup(448,858);
// by ID
G=gap.SmallGroup(448,858);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,9804,4911,172]);
// 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=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations