metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊3SD32, Q16⋊1D7, C8.6D14, C28.5D4, D56.2C2, C14.10D8, C56.4C22, C7⋊C16⋊3C2, (C7×Q16)⋊1C2, C2.6(D4⋊D7), C4.3(C7⋊D4), SmallGroup(224,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊SD32
G = < a,b,c | a7=b16=c2=1, bab-1=cac=a-1, cbc=b7 >
Smallest permutation representation of C7⋊SD32
►On 112 points
Generators in S112
(1 44 52 81 101 28 68)(2 69 29 102 82 53 45)(3 46 54 83 103 30 70)(4 71 31 104 84 55 47)(5 48 56 85 105 32 72)(6 73 17 106 86 57 33)(7 34 58 87 107 18 74)(8 75 19 108 88 59 35)(9 36 60 89 109 20 76)(10 77 21 110 90 61 37)(11 38 62 91 111 22 78)(12 79 23 112 92 63 39)(13 40 64 93 97 24 80)(14 65 25 98 94 49 41)(15 42 50 95 99 26 66)(16 67 27 100 96 51 43)
(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 55)(18 62)(19 53)(20 60)(21 51)(22 58)(23 49)(24 56)(25 63)(26 54)(27 61)(28 52)(29 59)(30 50)(31 57)(32 64)(33 71)(34 78)(35 69)(36 76)(37 67)(38 74)(39 65)(40 72)(41 79)(42 70)(43 77)(44 68)(45 75)(46 66)(47 73)(48 80)(81 101)(82 108)(83 99)(84 106)(85 97)(86 104)(87 111)(88 102)(89 109)(90 100)(91 107)(92 98)(93 105)(94 112)(95 103)(96 110)
G:=sub<Sym(112)| (1,44,52,81,101,28,68)(2,69,29,102,82,53,45)(3,46,54,83,103,30,70)(4,71,31,104,84,55,47)(5,48,56,85,105,32,72)(6,73,17,106,86,57,33)(7,34,58,87,107,18,74)(8,75,19,108,88,59,35)(9,36,60,89,109,20,76)(10,77,21,110,90,61,37)(11,38,62,91,111,22,78)(12,79,23,112,92,63,39)(13,40,64,93,97,24,80)(14,65,25,98,94,49,41)(15,42,50,95,99,26,66)(16,67,27,100,96,51,43), (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)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,71)(34,78)(35,69)(36,76)(37,67)(38,74)(39,65)(40,72)(41,79)(42,70)(43,77)(44,68)(45,75)(46,66)(47,73)(48,80)(81,101)(82,108)(83,99)(84,106)(85,97)(86,104)(87,111)(88,102)(89,109)(90,100)(91,107)(92,98)(93,105)(94,112)(95,103)(96,110)>;
G:=Group( (1,44,52,81,101,28,68)(2,69,29,102,82,53,45)(3,46,54,83,103,30,70)(4,71,31,104,84,55,47)(5,48,56,85,105,32,72)(6,73,17,106,86,57,33)(7,34,58,87,107,18,74)(8,75,19,108,88,59,35)(9,36,60,89,109,20,76)(10,77,21,110,90,61,37)(11,38,62,91,111,22,78)(12,79,23,112,92,63,39)(13,40,64,93,97,24,80)(14,65,25,98,94,49,41)(15,42,50,95,99,26,66)(16,67,27,100,96,51,43), (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)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,71)(34,78)(35,69)(36,76)(37,67)(38,74)(39,65)(40,72)(41,79)(42,70)(43,77)(44,68)(45,75)(46,66)(47,73)(48,80)(81,101)(82,108)(83,99)(84,106)(85,97)(86,104)(87,111)(88,102)(89,109)(90,100)(91,107)(92,98)(93,105)(94,112)(95,103)(96,110) );
G=PermutationGroup([[(1,44,52,81,101,28,68),(2,69,29,102,82,53,45),(3,46,54,83,103,30,70),(4,71,31,104,84,55,47),(5,48,56,85,105,32,72),(6,73,17,106,86,57,33),(7,34,58,87,107,18,74),(8,75,19,108,88,59,35),(9,36,60,89,109,20,76),(10,77,21,110,90,61,37),(11,38,62,91,111,22,78),(12,79,23,112,92,63,39),(13,40,64,93,97,24,80),(14,65,25,98,94,49,41),(15,42,50,95,99,26,66),(16,67,27,100,96,51,43)], [(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),(17,55),(18,62),(19,53),(20,60),(21,51),(22,58),(23,49),(24,56),(25,63),(26,54),(27,61),(28,52),(29,59),(30,50),(31,57),(32,64),(33,71),(34,78),(35,69),(36,76),(37,67),(38,74),(39,65),(40,72),(41,79),(42,70),(43,77),(44,68),(45,75),(46,66),(47,73),(48,80),(81,101),(82,108),(83,99),(84,106),(85,97),(86,104),(87,111),(88,102),(89,109),(90,100),(91,107),(92,98),(93,105),(94,112),(95,103),(96,110)]])
C7⋊SD32 is a maximal subgroup of
D7×SD32 D112⋊C2 Q32⋊D7 Q32⋊3D7 Q16.D14 Q16⋊D14 C56.30C23
C7⋊SD32 is a maximal quotient of C8.5Dic14 C14.D16 C14.Q32
32 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 56 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | SD32 | C7⋊D4 | D4⋊D7 | C7⋊SD32 |
| kernel | C7⋊SD32 | C7⋊C16 | D56 | C7×Q16 | C28 | Q16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 3 | 6 |
Matrix representation of C7⋊SD32 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 9 |
| 23 | 22 | 0 | 0 |
| 68 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 112 |
| 1 | 0 | 0 | 0 |
| 43 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,9],[23,68,0,0,22,65,0,0,0,0,1,9,0,0,0,112],[1,43,0,0,0,112,0,0,0,0,1,9,0,0,0,112] >;
C7⋊SD32 in GAP, Magma, Sage, TeX
C_7\rtimes {\rm SD}_{32} % in TeX
G:=Group("C7:SD32"); // GroupNames label
G:=SmallGroup(224,34);
// by ID
G=gap.SmallGroup(224,34);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,103,218,116,122,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^7>;
// generators/relations
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