direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×SD32, D8.C14, C16⋊2C14, C112⋊6C2, Q16⋊1C14, C14.16D8, C28.37D4, C56.25C22, C2.4(C7×D8), C4.2(C7×D4), C8.3(C2×C14), (C7×Q16)⋊5C2, (C7×D8).2C2, SmallGroup(224,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×SD32
G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C7×SD32
►On 112 points
Generators in S112
(1 94 67 58 39 22 107)(2 95 68 59 40 23 108)(3 96 69 60 41 24 109)(4 81 70 61 42 25 110)(5 82 71 62 43 26 111)(6 83 72 63 44 27 112)(7 84 73 64 45 28 97)(8 85 74 49 46 29 98)(9 86 75 50 47 30 99)(10 87 76 51 48 31 100)(11 88 77 52 33 32 101)(12 89 78 53 34 17 102)(13 90 79 54 35 18 103)(14 91 80 55 36 19 104)(15 92 65 56 37 20 105)(16 93 66 57 38 21 106)
(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)(17 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 59)(51 57)(52 64)(53 55)(54 62)(56 60)(61 63)(65 69)(66 76)(68 74)(70 72)(71 79)(73 77)(78 80)(81 83)(82 90)(84 88)(85 95)(87 93)(89 91)(92 96)(97 101)(98 108)(100 106)(102 104)(103 111)(105 109)(110 112)
G:=sub<Sym(112)| (1,94,67,58,39,22,107)(2,95,68,59,40,23,108)(3,96,69,60,41,24,109)(4,81,70,61,42,25,110)(5,82,71,62,43,26,111)(6,83,72,63,44,27,112)(7,84,73,64,45,28,97)(8,85,74,49,46,29,98)(9,86,75,50,47,30,99)(10,87,76,51,48,31,100)(11,88,77,52,33,32,101)(12,89,78,53,34,17,102)(13,90,79,54,35,18,103)(14,91,80,55,36,19,104)(15,92,65,56,37,20,105)(16,93,66,57,38,21,106), (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)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,59)(51,57)(52,64)(53,55)(54,62)(56,60)(61,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96)(97,101)(98,108)(100,106)(102,104)(103,111)(105,109)(110,112)>;
G:=Group( (1,94,67,58,39,22,107)(2,95,68,59,40,23,108)(3,96,69,60,41,24,109)(4,81,70,61,42,25,110)(5,82,71,62,43,26,111)(6,83,72,63,44,27,112)(7,84,73,64,45,28,97)(8,85,74,49,46,29,98)(9,86,75,50,47,30,99)(10,87,76,51,48,31,100)(11,88,77,52,33,32,101)(12,89,78,53,34,17,102)(13,90,79,54,35,18,103)(14,91,80,55,36,19,104)(15,92,65,56,37,20,105)(16,93,66,57,38,21,106), (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)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,59)(51,57)(52,64)(53,55)(54,62)(56,60)(61,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96)(97,101)(98,108)(100,106)(102,104)(103,111)(105,109)(110,112) );
G=PermutationGroup([[(1,94,67,58,39,22,107),(2,95,68,59,40,23,108),(3,96,69,60,41,24,109),(4,81,70,61,42,25,110),(5,82,71,62,43,26,111),(6,83,72,63,44,27,112),(7,84,73,64,45,28,97),(8,85,74,49,46,29,98),(9,86,75,50,47,30,99),(10,87,76,51,48,31,100),(11,88,77,52,33,32,101),(12,89,78,53,34,17,102),(13,90,79,54,35,18,103),(14,91,80,55,36,19,104),(15,92,65,56,37,20,105),(16,93,66,57,38,21,106)], [(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),(17,19),(18,26),(20,24),(21,31),(23,29),(25,27),(28,32),(33,45),(34,36),(35,43),(37,41),(38,48),(40,46),(42,44),(49,59),(51,57),(52,64),(53,55),(54,62),(56,60),(61,63),(65,69),(66,76),(68,74),(70,72),(71,79),(73,77),(78,80),(81,83),(82,90),(84,88),(85,95),(87,93),(89,91),(92,96),(97,101),(98,108),(100,106),(102,104),(103,111),(105,109),(110,112)]])
C7×SD32 is a maximal subgroup of
D112⋊C2 SD32⋊D7 SD32⋊3D7
77 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 8 | 2 | 8 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D8 | SD32 | C7×D4 | C7×D8 | C7×SD32 |
| kernel | C7×SD32 | C112 | C7×D8 | C7×Q16 | SD32 | C16 | D8 | Q16 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 2 | 4 | 6 | 12 | 24 |
Matrix representation of C7×SD32 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 53 | 69 |
| 0 | 0 | 44 | 53 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,1,0,0,0,0,1],[82,82,0,0,31,82,0,0,0,0,53,44,0,0,69,53],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112] >;
C7×SD32 in GAP, Magma, Sage, TeX
C_7\times {\rm SD}_{32} % in TeX
G:=Group("C7xSD32"); // GroupNames label
G:=SmallGroup(224,61);
// by ID
G=gap.SmallGroup(224,61);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,672,361,2019,1017,165,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
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