direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C16⋊C22, D16⋊2C14, C28.65D8, C56.53D4, C112⋊6C22, SD32⋊1C14, M5(2)⋊1C14, C56.76C23, C16⋊(C2×C14), C8.3(C7×D4), C4○D8⋊2C14, D8⋊2(C2×C14), (C7×D16)⋊6C2, C4.14(C7×D8), (C14×D8)⋊24C2, (C2×D8)⋊10C14, Q16⋊2(C2×C14), (C7×SD32)⋊5C2, C14.88(C2×D8), (C2×C14).27D8, C2.16(C14×D8), C4.11(D4×C14), C22.5(C7×D8), C28.318(C2×D4), (C2×C28).346D4, (C7×D8)⋊18C22, (C7×M5(2))⋊3C2, C8.7(C22×C14), (C7×Q16)⋊16C22, (C2×C56).278C22, (C7×C4○D8)⋊9C2, (C2×C4).47(C7×D4), (C2×C8).30(C2×C14), SmallGroup(448,917)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C16⋊C22
G = < a,b,c,d | a7=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >
Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C16, C2×C8, D8, D8, D8, SD16, Q16, C2×D4, C4○D4, C28, C28, C2×C14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C16⋊C22, C112, C2×C56, C7×D8, C7×D8, C7×D8, C7×SD16, C7×Q16, D4×C14, C7×C4○D4, C7×M5(2), C7×D16, C7×SD32, C14×D8, C7×C4○D8, C7×C16⋊C22
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C2×D8, C7×D4, C22×C14, C16⋊C22, C7×D8, D4×C14, C14×D8, C7×C16⋊C22
►On 112 points
Generators in S112
(1 71 39 50 18 92 109)(2 72 40 51 19 93 110)(3 73 41 52 20 94 111)(4 74 42 53 21 95 112)(5 75 43 54 22 96 97)(6 76 44 55 23 81 98)(7 77 45 56 24 82 99)(8 78 46 57 25 83 100)(9 79 47 58 26 84 101)(10 80 48 59 27 85 102)(11 65 33 60 28 86 103)(12 66 34 61 29 87 104)(13 67 35 62 30 88 105)(14 68 36 63 31 89 106)(15 69 37 64 32 90 107)(16 70 38 49 17 91 108)
(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 11)(3 9)(4 16)(5 7)(6 14)(8 12)(13 15)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 39)(34 46)(35 37)(36 44)(38 42)(41 47)(43 45)(49 53)(50 60)(52 58)(54 56)(55 63)(57 61)(62 64)(65 71)(66 78)(67 69)(68 76)(70 74)(73 79)(75 77)(81 89)(82 96)(83 87)(84 94)(86 92)(88 90)(91 95)(97 99)(98 106)(100 104)(101 111)(103 109)(105 107)(108 112)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)
G:=sub<Sym(112)| (1,71,39,50,18,92,109)(2,72,40,51,19,93,110)(3,73,41,52,20,94,111)(4,74,42,53,21,95,112)(5,75,43,54,22,96,97)(6,76,44,55,23,81,98)(7,77,45,56,24,82,99)(8,78,46,57,25,83,100)(9,79,47,58,26,84,101)(10,80,48,59,27,85,102)(11,65,33,60,28,86,103)(12,66,34,61,29,87,104)(13,67,35,62,30,88,105)(14,68,36,63,31,89,106)(15,69,37,64,32,90,107)(16,70,38,49,17,91,108), (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), (1,11)(3,9)(4,16)(5,7)(6,14)(8,12)(13,15)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,39)(34,46)(35,37)(36,44)(38,42)(41,47)(43,45)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,71)(66,78)(67,69)(68,76)(70,74)(73,79)(75,77)(81,89)(82,96)(83,87)(84,94)(86,92)(88,90)(91,95)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112)>;
G:=Group( (1,71,39,50,18,92,109)(2,72,40,51,19,93,110)(3,73,41,52,20,94,111)(4,74,42,53,21,95,112)(5,75,43,54,22,96,97)(6,76,44,55,23,81,98)(7,77,45,56,24,82,99)(8,78,46,57,25,83,100)(9,79,47,58,26,84,101)(10,80,48,59,27,85,102)(11,65,33,60,28,86,103)(12,66,34,61,29,87,104)(13,67,35,62,30,88,105)(14,68,36,63,31,89,106)(15,69,37,64,32,90,107)(16,70,38,49,17,91,108), (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), (1,11)(3,9)(4,16)(5,7)(6,14)(8,12)(13,15)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,39)(34,46)(35,37)(36,44)(38,42)(41,47)(43,45)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,71)(66,78)(67,69)(68,76)(70,74)(73,79)(75,77)(81,89)(82,96)(83,87)(84,94)(86,92)(88,90)(91,95)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112) );
G=PermutationGroup([[(1,71,39,50,18,92,109),(2,72,40,51,19,93,110),(3,73,41,52,20,94,111),(4,74,42,53,21,95,112),(5,75,43,54,22,96,97),(6,76,44,55,23,81,98),(7,77,45,56,24,82,99),(8,78,46,57,25,83,100),(9,79,47,58,26,84,101),(10,80,48,59,27,85,102),(11,65,33,60,28,86,103),(12,66,34,61,29,87,104),(13,67,35,62,30,88,105),(14,68,36,63,31,89,106),(15,69,37,64,32,90,107),(16,70,38,49,17,91,108)], [(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)], [(1,11),(3,9),(4,16),(5,7),(6,14),(8,12),(13,15),(17,21),(18,28),(20,26),(22,24),(23,31),(25,29),(30,32),(33,39),(34,46),(35,37),(36,44),(38,42),(41,47),(43,45),(49,53),(50,60),(52,58),(54,56),(55,63),(57,61),(62,64),(65,71),(66,78),(67,69),(68,76),(70,74),(73,79),(75,77),(81,89),(82,96),(83,87),(84,94),(86,92),(88,90),(91,95),(97,99),(98,106),(100,104),(101,111),(103,109),(105,107),(108,112)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(66,74),(68,76),(70,78),(72,80),(81,89),(83,91),(85,93),(87,95),(98,106),(100,108),(102,110),(104,112)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 8C | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14AD | 16A | 16B | 16C | 16D | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 2 | 2 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
112 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 | D8 | D8 | C7×D4 | C7×D4 | C7×D8 | C7×D8 | C16⋊C22 | C7×C16⋊C22 |
| kernel | C7×C16⋊C22 | C7×M5(2) | C7×D16 | C7×SD32 | C14×D8 | C7×C4○D8 | C16⋊C22 | M5(2) | D16 | SD32 | C2×D8 | C4○D8 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C16⋊C22 ►in GL6(𝔽113)
| 49 | 0 | 0 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 83 | 29 | 0 | 0 | 0 | 0 |
| 4 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 59 | 0 | 51 |
| 0 | 0 | 70 | 59 | 62 | 51 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 85 | 53 | 59 | 97 |
| 112 | 98 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 82 | 31 | 0 | 0 |
| 0 | 0 | 31 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 74 | 29 | 1 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 46 | 42 | 0 | 112 |
G:=sub<GL(6,GF(113))| [49,0,0,0,0,0,0,49,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[83,4,0,0,0,0,29,30,0,0,0,0,0,0,70,70,112,85,0,0,59,59,0,53,0,0,0,62,0,59,0,0,51,51,0,97],[112,0,0,0,0,0,98,1,0,0,0,0,0,0,82,31,0,74,0,0,31,31,0,29,0,0,0,0,1,1,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,46,0,0,0,1,0,42,0,0,0,0,112,0,0,0,0,0,0,112] >;
C7×C16⋊C22 in GAP, Magma, Sage, TeX
C_7\times C_{16}\rtimes C_2^2 % in TeX
G:=Group("C7xC16:C2^2"); // GroupNames label
G:=SmallGroup(448,917);
// by ID
G=gap.SmallGroup(448,917);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,4790,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations