direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×SD32, C16⋊5D14, Q16⋊1D14, D8.2D14, C112⋊5C22, D14.13D8, Dic7.4D8, C56.16C23, D56.2C22, Dic28⋊5C22, (D7×D8).C2, C7⋊C8.13D4, C4.4(D4×D7), C7⋊2(C2×SD32), (D7×C16)⋊4C2, C7⋊C16⋊6C22, D8.D7⋊3C2, (D7×Q16)⋊3C2, C2.19(D7×D8), C112⋊C2⋊5C2, (C7×SD32)⋊3C2, (C4×D7).20D4, C28.10(C2×D4), C14.35(C2×D8), C7⋊SD32⋊1C2, (C7×Q16)⋊4C22, (C7×D8).2C22, C8.22(C22×D7), (C8×D7).11C22, SmallGroup(448,447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×SD32
G = < a,b,c,d | a7=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
Subgroups: 672 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C23, D7, D7, C14, C14, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C16, SD32, SD32, C2×D8, C2×Q16, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C2×SD32, C7⋊C16, C112, C8×D7, D56, Dic28, D4⋊D7, C7⋊Q16, C7×D8, C7×Q16, D4×D7, Q8×D7, D7×C16, C112⋊C2, D8.D7, C7⋊SD32, C7×SD32, D7×D8, D7×Q16, D7×SD32
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, SD32, C2×D8, C22×D7, C2×SD32, D4×D7, D7×D8, D7×SD32
►On 112 points
Generators in S112
(1 71 97 50 42 90 25)(2 72 98 51 43 91 26)(3 73 99 52 44 92 27)(4 74 100 53 45 93 28)(5 75 101 54 46 94 29)(6 76 102 55 47 95 30)(7 77 103 56 48 96 31)(8 78 104 57 33 81 32)(9 79 105 58 34 82 17)(10 80 106 59 35 83 18)(11 65 107 60 36 84 19)(12 66 108 61 37 85 20)(13 67 109 62 38 86 21)(14 68 110 63 39 87 22)(15 69 111 64 40 88 23)(16 70 112 49 41 89 24)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 81)(79 82)(80 83)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)(49 59)(51 57)(52 64)(53 55)(54 62)(56 60)(61 63)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)(81 91)(83 89)(84 96)(85 87)(86 94)(88 92)(93 95)(98 104)(99 111)(100 102)(101 109)(103 107)(106 112)(108 110)
G:=sub<Sym(112)| (1,71,97,50,42,90,25)(2,72,98,51,43,91,26)(3,73,99,52,44,92,27)(4,74,100,53,45,93,28)(5,75,101,54,46,94,29)(6,76,102,55,47,95,30)(7,77,103,56,48,96,31)(8,78,104,57,33,81,32)(9,79,105,58,34,82,17)(10,80,106,59,35,83,18)(11,65,107,60,36,84,19)(12,66,108,61,37,85,20)(13,67,109,62,38,86,21)(14,68,110,63,39,87,22)(15,69,111,64,40,88,23)(16,70,112,49,41,89,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,81)(79,82)(80,83), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,59)(51,57)(52,64)(53,55)(54,62)(56,60)(61,63)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(98,104)(99,111)(100,102)(101,109)(103,107)(106,112)(108,110)>;
G:=Group( (1,71,97,50,42,90,25)(2,72,98,51,43,91,26)(3,73,99,52,44,92,27)(4,74,100,53,45,93,28)(5,75,101,54,46,94,29)(6,76,102,55,47,95,30)(7,77,103,56,48,96,31)(8,78,104,57,33,81,32)(9,79,105,58,34,82,17)(10,80,106,59,35,83,18)(11,65,107,60,36,84,19)(12,66,108,61,37,85,20)(13,67,109,62,38,86,21)(14,68,110,63,39,87,22)(15,69,111,64,40,88,23)(16,70,112,49,41,89,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,81)(79,82)(80,83), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,59)(51,57)(52,64)(53,55)(54,62)(56,60)(61,63)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(98,104)(99,111)(100,102)(101,109)(103,107)(106,112)(108,110) );
G=PermutationGroup([[(1,71,97,50,42,90,25),(2,72,98,51,43,91,26),(3,73,99,52,44,92,27),(4,74,100,53,45,93,28),(5,75,101,54,46,94,29),(6,76,102,55,47,95,30),(7,77,103,56,48,96,31),(8,78,104,57,33,81,32),(9,79,105,58,34,82,17),(10,80,106,59,35,83,18),(11,65,107,60,36,84,19),(12,66,108,61,37,85,20),(13,67,109,62,38,86,21),(14,68,110,63,39,87,22),(15,69,111,64,40,88,23),(16,70,112,49,41,89,24)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,81),(79,82),(80,83)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(18,24),(19,31),(20,22),(21,29),(23,27),(26,32),(28,30),(33,43),(35,41),(36,48),(37,39),(38,46),(40,44),(45,47),(49,59),(51,57),(52,64),(53,55),(54,62),(56,60),(61,63),(65,77),(66,68),(67,75),(69,73),(70,80),(72,78),(74,76),(81,91),(83,89),(84,96),(85,87),(86,94),(88,92),(93,95),(98,104),(99,111),(100,102),(101,109),(103,107),(106,112),(108,110)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 28A | 28B | 28C | 28D | 28E | 28F | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 7 | 7 | 8 | 56 | 2 | 8 | 14 | 56 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 16 | 16 | 16 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | D14 | SD32 | D4×D7 | D7×D8 | D7×SD32 |
| kernel | D7×SD32 | D7×C16 | C112⋊C2 | D8.D7 | C7⋊SD32 | C7×SD32 | D7×D8 | D7×Q16 | C7⋊C8 | C4×D7 | SD32 | Dic7 | D14 | C16 | D8 | Q16 | D7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 8 | 3 | 6 | 12 |
Matrix representation of D7×SD32 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 1 |
| 0 | 0 | 111 | 104 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 79 |
| 0 | 0 | 24 | 79 |
| 104 | 44 | 0 | 0 |
| 25 | 16 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,88,111,0,0,1,104],[1,0,0,0,0,1,0,0,0,0,34,24,0,0,79,79],[104,25,0,0,44,16,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,1,112,0,0,0,0,1,0,0,0,0,1] >;
D7×SD32 in GAP, Magma, Sage, TeX
D_7\times {\rm SD}_{32} % in TeX
G:=Group("D7xSD32"); // GroupNames label
G:=SmallGroup(448,447);
// by ID
G=gap.SmallGroup(448,447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,135,184,346,185,192,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations