direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4○D8, D8⋊3C14, Q16⋊3C14, C28.69D4, SD16⋊3C14, C28.47C23, C56.28C22, (C2×C8)⋊4C14, (C7×D8)⋊7C2, (C2×C56)⋊12C2, C4○D4⋊1C14, C8.6(C2×C14), (C7×Q16)⋊7C2, C4.20(C7×D4), (C7×SD16)⋊7C2, D4.2(C2×C14), C2.14(D4×C14), C14.77(C2×D4), (C2×C14).11D4, Q8.2(C2×C14), C22.1(C7×D4), C4.4(C22×C14), (C7×D4).12C22, (C7×Q8).13C22, (C2×C28).132C22, (C7×C4○D4)⋊6C2, (C2×C4).28(C2×C14), SmallGroup(224,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4○D8
G = < a,b,c,d | a7=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C4○D8, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C7×C4○D8
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C4○D8, C7×D4, C22×C14, D4×C14, C7×C4○D8
►On 112 points
Generators in S112
(1 73 43 23 93 65 35)(2 74 44 24 94 66 36)(3 75 45 17 95 67 37)(4 76 46 18 96 68 38)(5 77 47 19 89 69 39)(6 78 48 20 90 70 40)(7 79 41 21 91 71 33)(8 80 42 22 92 72 34)(9 112 84 54 26 104 57)(10 105 85 55 27 97 58)(11 106 86 56 28 98 59)(12 107 87 49 29 99 60)(13 108 88 50 30 100 61)(14 109 81 51 31 101 62)(15 110 82 52 32 102 63)(16 111 83 53 25 103 64)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 107 21 111)(18 108 22 112)(19 109 23 105)(20 110 24 106)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)(49 71 53 67)(50 72 54 68)(51 65 55 69)(52 66 56 70)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 93 85 89)(82 94 86 90)(83 95 87 91)(84 96 88 92)
(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 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 53)(50 52)(54 56)(57 59)(60 64)(61 63)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(82 88)(83 87)(84 86)(90 96)(91 95)(92 94)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)
G:=sub<Sym(112)| (1,73,43,23,93,65,35)(2,74,44,24,94,66,36)(3,75,45,17,95,67,37)(4,76,46,18,96,68,38)(5,77,47,19,89,69,39)(6,78,48,20,90,70,40)(7,79,41,21,91,71,33)(8,80,42,22,92,72,34)(9,112,84,54,26,104,57)(10,105,85,55,27,97,58)(11,106,86,56,28,98,59)(12,107,87,49,29,99,60)(13,108,88,50,30,100,61)(14,109,81,51,31,101,62)(15,110,82,52,32,102,63)(16,111,83,53,25,103,64), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36)(49,71,53,67)(50,72,54,68)(51,65,55,69)(52,66,56,70)(57,76,61,80)(58,77,62,73)(59,78,63,74)(60,79,64,75)(81,93,85,89)(82,94,86,90)(83,95,87,91)(84,96,88,92), (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)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,53)(50,52)(54,56)(57,59)(60,64)(61,63)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(82,88)(83,87)(84,86)(90,96)(91,95)(92,94)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)>;
G:=Group( (1,73,43,23,93,65,35)(2,74,44,24,94,66,36)(3,75,45,17,95,67,37)(4,76,46,18,96,68,38)(5,77,47,19,89,69,39)(6,78,48,20,90,70,40)(7,79,41,21,91,71,33)(8,80,42,22,92,72,34)(9,112,84,54,26,104,57)(10,105,85,55,27,97,58)(11,106,86,56,28,98,59)(12,107,87,49,29,99,60)(13,108,88,50,30,100,61)(14,109,81,51,31,101,62)(15,110,82,52,32,102,63)(16,111,83,53,25,103,64), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36)(49,71,53,67)(50,72,54,68)(51,65,55,69)(52,66,56,70)(57,76,61,80)(58,77,62,73)(59,78,63,74)(60,79,64,75)(81,93,85,89)(82,94,86,90)(83,95,87,91)(84,96,88,92), (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)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,53)(50,52)(54,56)(57,59)(60,64)(61,63)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(82,88)(83,87)(84,86)(90,96)(91,95)(92,94)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110) );
G=PermutationGroup([[(1,73,43,23,93,65,35),(2,74,44,24,94,66,36),(3,75,45,17,95,67,37),(4,76,46,18,96,68,38),(5,77,47,19,89,69,39),(6,78,48,20,90,70,40),(7,79,41,21,91,71,33),(8,80,42,22,92,72,34),(9,112,84,54,26,104,57),(10,105,85,55,27,97,58),(11,106,86,56,28,98,59),(12,107,87,49,29,99,60),(13,108,88,50,30,100,61),(14,109,81,51,31,101,62),(15,110,82,52,32,102,63),(16,111,83,53,25,103,64)], [(1,97,5,101),(2,98,6,102),(3,99,7,103),(4,100,8,104),(9,46,13,42),(10,47,14,43),(11,48,15,44),(12,41,16,45),(17,107,21,111),(18,108,22,112),(19,109,23,105),(20,110,24,106),(25,37,29,33),(26,38,30,34),(27,39,31,35),(28,40,32,36),(49,71,53,67),(50,72,54,68),(51,65,55,69),(52,66,56,70),(57,76,61,80),(58,77,62,73),(59,78,63,74),(60,79,64,75),(81,93,85,89),(82,94,86,90),(83,95,87,91),(84,96,88,92)], [(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),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24),(25,29),(26,28),(30,32),(33,37),(34,36),(38,40),(41,45),(42,44),(46,48),(49,53),(50,52),(54,56),(57,59),(60,64),(61,63),(66,72),(67,71),(68,70),(74,80),(75,79),(76,78),(82,88),(83,87),(84,86),(90,96),(91,95),(92,94),(98,104),(99,103),(100,102),(106,112),(107,111),(108,110)]])
C7×C4○D8 is a maximal subgroup of
D8⋊2Dic7 C28.58D8 Q16⋊D14 C56.30C23 C56.31C23 D8⋊5Dic7 D8⋊4Dic7 D8⋊10D14 D8⋊15D14 D8⋊11D14 D8.10D14
C7×C4○D8 is a maximal quotient of
D8×C28 SD16×C28 Q16×C28
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D8 | C7×D4 | C7×D4 | C7×C4○D8 |
| kernel | C7×C4○D8 | C2×C56 | C7×D8 | C7×SD16 | C7×Q16 | C7×C4○D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C28 | C2×C14 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 6 | 6 | 6 | 12 | 6 | 12 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C7×C4○D8 ►in GL2(𝔽113) generated by
| 16 | 0 |
| 0 | 16 |
| 98 | 0 |
| 0 | 98 |
| 51 | 82 |
| 62 | 0 |
| 1 | 1 |
| 0 | 112 |
G:=sub<GL(2,GF(113))| [16,0,0,16],[98,0,0,98],[51,62,82,0],[1,0,1,112] >;
C7×C4○D8 in GAP, Magma, Sage, TeX
C_7\times C_4\circ D_8
% in TeX
G:=Group("C7xC4oD8"); // GroupNames label
G:=SmallGroup(224,170);
// by ID
G=gap.SmallGroup(224,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,518,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations