direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.31D4, C4⋊C4⋊2C28, (C2×Q8)⋊1C28, (Q8×C14)⋊3C4, C14.23C4≀C2, (C2×C28).444D4, C22⋊C8.2C14, (C2×C14).11Q16, C23.30(C7×D4), C22⋊Q8.1C14, C22.2(C7×Q16), (C2×C14).23SD16, C22.2(C7×SD16), C14.30(C23⋊C4), (C22×C14).150D4, C14.15(Q8⋊C4), C2.C42.5C14, (C22×C28).385C22, (C7×C4⋊C4)⋊4C4, C2.5(C7×C4≀C2), (C2×C4).8(C2×C28), (C2×C4).96(C7×D4), C2.5(C7×C23⋊C4), (C7×C22⋊C8).4C2, C2.3(C7×Q8⋊C4), (C2×C28).175(C2×C4), (C7×C22⋊Q8).11C2, (C22×C4).15(C2×C14), C22.36(C7×C22⋊C4), (C2×C14).123(C22⋊C4), (C7×C2.C42).24C2, SmallGroup(448,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.31D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >
Subgroups: 162 in 80 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C22⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C22×C14, C23.31D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C22×C28, Q8×C14, C7×C2.C42, C7×C22⋊C8, C7×C22⋊Q8, C7×C23.31D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, SD16, Q16, C28, C2×C14, C23⋊C4, Q8⋊C4, C4≀C2, C2×C28, C7×D4, C23.31D4, C7×C22⋊C4, C7×SD16, C7×Q16, C7×C23⋊C4, C7×Q8⋊C4, C7×C4≀C2, C7×C23.31D4
►On 112 points
Generators in S112
(1 57 109 55 101 47 93)(2 58 110 56 102 48 94)(3 59 111 49 103 41 95)(4 60 112 50 104 42 96)(5 61 105 51 97 43 89)(6 62 106 52 98 44 90)(7 63 107 53 99 45 91)(8 64 108 54 100 46 92)(9 36 82 28 74 20 66)(10 37 83 29 75 21 67)(11 38 84 30 76 22 68)(12 39 85 31 77 23 69)(13 40 86 32 78 24 70)(14 33 87 25 79 17 71)(15 34 88 26 80 18 72)(16 35 81 27 73 19 65)
(2 70)(4 72)(6 66)(8 68)(9 62)(11 64)(13 58)(15 60)(18 96)(20 90)(22 92)(24 94)(26 104)(28 98)(30 100)(32 102)(34 112)(36 106)(38 108)(40 110)(42 80)(44 74)(46 76)(48 78)(50 88)(52 82)(54 84)(56 86)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 89)(24 90)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 97)(32 98)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 105)(40 106)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(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)
(1 5)(2 8 70 68)(3 71)(4 66 72 6)(7 67)(9 15 62 60)(10 63)(11 58 64 13)(12 16)(14 59)(17 95)(18 90 96 20)(19 23)(21 91)(22 94 92 24)(25 103)(26 98 104 28)(27 31)(29 99)(30 102 100 32)(33 111)(34 106 112 36)(35 39)(37 107)(38 110 108 40)(41 79)(42 74 80 44)(43 47)(45 75)(46 78 76 48)(49 87)(50 82 88 52)(51 55)(53 83)(54 86 84 56)(57 61)(65 69)(73 77)(81 85)(89 93)(97 101)(105 109)
G:=sub<Sym(112)| (1,57,109,55,101,47,93)(2,58,110,56,102,48,94)(3,59,111,49,103,41,95)(4,60,112,50,104,42,96)(5,61,105,51,97,43,89)(6,62,106,52,98,44,90)(7,63,107,53,99,45,91)(8,64,108,54,100,46,92)(9,36,82,28,74,20,66)(10,37,83,29,75,21,67)(11,38,84,30,76,22,68)(12,39,85,31,77,23,69)(13,40,86,32,78,24,70)(14,33,87,25,79,17,71)(15,34,88,26,80,18,72)(16,35,81,27,73,19,65), (2,70)(4,72)(6,66)(8,68)(9,62)(11,64)(13,58)(15,60)(18,96)(20,90)(22,92)(24,94)(26,104)(28,98)(30,100)(32,102)(34,112)(36,106)(38,108)(40,110)(42,80)(44,74)(46,76)(48,78)(50,88)(52,82)(54,84)(56,86), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,89)(24,90)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,97)(32,98)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,105)(40,106)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (1,5)(2,8,70,68)(3,71)(4,66,72,6)(7,67)(9,15,62,60)(10,63)(11,58,64,13)(12,16)(14,59)(17,95)(18,90,96,20)(19,23)(21,91)(22,94,92,24)(25,103)(26,98,104,28)(27,31)(29,99)(30,102,100,32)(33,111)(34,106,112,36)(35,39)(37,107)(38,110,108,40)(41,79)(42,74,80,44)(43,47)(45,75)(46,78,76,48)(49,87)(50,82,88,52)(51,55)(53,83)(54,86,84,56)(57,61)(65,69)(73,77)(81,85)(89,93)(97,101)(105,109)>;
G:=Group( (1,57,109,55,101,47,93)(2,58,110,56,102,48,94)(3,59,111,49,103,41,95)(4,60,112,50,104,42,96)(5,61,105,51,97,43,89)(6,62,106,52,98,44,90)(7,63,107,53,99,45,91)(8,64,108,54,100,46,92)(9,36,82,28,74,20,66)(10,37,83,29,75,21,67)(11,38,84,30,76,22,68)(12,39,85,31,77,23,69)(13,40,86,32,78,24,70)(14,33,87,25,79,17,71)(15,34,88,26,80,18,72)(16,35,81,27,73,19,65), (2,70)(4,72)(6,66)(8,68)(9,62)(11,64)(13,58)(15,60)(18,96)(20,90)(22,92)(24,94)(26,104)(28,98)(30,100)(32,102)(34,112)(36,106)(38,108)(40,110)(42,80)(44,74)(46,76)(48,78)(50,88)(52,82)(54,84)(56,86), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,89)(24,90)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,97)(32,98)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,105)(40,106)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (1,5)(2,8,70,68)(3,71)(4,66,72,6)(7,67)(9,15,62,60)(10,63)(11,58,64,13)(12,16)(14,59)(17,95)(18,90,96,20)(19,23)(21,91)(22,94,92,24)(25,103)(26,98,104,28)(27,31)(29,99)(30,102,100,32)(33,111)(34,106,112,36)(35,39)(37,107)(38,110,108,40)(41,79)(42,74,80,44)(43,47)(45,75)(46,78,76,48)(49,87)(50,82,88,52)(51,55)(53,83)(54,86,84,56)(57,61)(65,69)(73,77)(81,85)(89,93)(97,101)(105,109) );
G=PermutationGroup([[(1,57,109,55,101,47,93),(2,58,110,56,102,48,94),(3,59,111,49,103,41,95),(4,60,112,50,104,42,96),(5,61,105,51,97,43,89),(6,62,106,52,98,44,90),(7,63,107,53,99,45,91),(8,64,108,54,100,46,92),(9,36,82,28,74,20,66),(10,37,83,29,75,21,67),(11,38,84,30,76,22,68),(12,39,85,31,77,23,69),(13,40,86,32,78,24,70),(14,33,87,25,79,17,71),(15,34,88,26,80,18,72),(16,35,81,27,73,19,65)], [(2,70),(4,72),(6,66),(8,68),(9,62),(11,64),(13,58),(15,60),(18,96),(20,90),(22,92),(24,94),(26,104),(28,98),(30,100),(32,102),(34,112),(36,106),(38,108),(40,110),(42,80),(44,74),(46,76),(48,78),(50,88),(52,82),(54,84),(56,86)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,89),(24,90),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,97),(32,98),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,105),(40,106),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,81),(56,82)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(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)], [(1,5),(2,8,70,68),(3,71),(4,66,72,6),(7,67),(9,15,62,60),(10,63),(11,58,64,13),(12,16),(14,59),(17,95),(18,90,96,20),(19,23),(21,91),(22,94,92,24),(25,103),(26,98,104,28),(27,31),(29,99),(30,102,100,32),(33,111),(34,106,112,36),(35,39),(37,107),(38,110,108,40),(41,79),(42,74,80,44),(43,47),(45,75),(46,78,76,48),(49,87),(50,82,88,52),(51,55),(53,83),(54,86,84,56),(57,61),(65,69),(73,77),(81,85),(89,93),(97,101),(105,109)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28AP | 28AQ | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | D4 | D4 | SD16 | Q16 | C4≀C2 | C7×D4 | C7×D4 | C7×SD16 | C7×Q16 | C7×C4≀C2 | C23⋊C4 | C7×C23⋊C4 |
| kernel | C7×C23.31D4 | C7×C2.C42 | C7×C22⋊C8 | C7×C22⋊Q8 | C7×C4⋊C4 | Q8×C14 | C23.31D4 | C2.C42 | C22⋊C8 | C22⋊Q8 | C4⋊C4 | C2×Q8 | C2×C28 | C22×C14 | C2×C14 | C2×C14 | C14 | C2×C4 | C23 | C22 | C22 | C2 | C14 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 24 | 1 | 6 |
Matrix representation of C7×C23.31D4 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 1 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 15 | 30 | 0 | 0 |
| 106 | 98 | 0 | 0 |
| 0 | 0 | 13 | 100 |
| 0 | 0 | 13 | 13 |
| 112 | 0 | 0 | 0 |
| 106 | 98 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,106,0,0,0,0,106],[1,112,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[15,106,0,0,30,98,0,0,0,0,13,13,0,0,100,13],[112,106,0,0,0,98,0,0,0,0,112,0,0,0,0,1] >;
C7×C23.31D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{31}D_4 % in TeX
G:=Group("C7xC2^3.31D4"); // GroupNames label
G:=SmallGroup(448,132);
// by ID
G=gap.SmallGroup(448,132);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,3923,3538,248,6871]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;
// generators/relations