direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7xC8:C22, D8:2C14, C56:7C22, C28.63D4, SD16:1C14, M4(2):1C14, C28.48C23, C8:(C2xC14), (C7xD8):6C2, C4oD4:2C14, (C2xD4):5C14, D4:2(C2xC14), Q8:2(C2xC14), C4.14(C7xD4), (D4xC14):14C2, (C7xSD16):5C2, C2.15(D4xC14), C14.78(C2xD4), (C2xC14).24D4, C22.5(C7xD4), (C7xD4):11C22, (C7xM4(2)):5C2, C4.5(C22xC14), (C7xQ8):10C22, (C2xC28).69C22, (C7xC4oD4):7C2, (C2xC4).10(C2xC14), SmallGroup(224,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xC8:C22
G = < a,b,c,d | a7=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, D4, D4, Q8, C23, C14, C14, M4(2), D8, SD16, C2xD4, C4oD4, C28, C28, C2xC14, C2xC14, C8:C22, C56, C2xC28, C2xC28, C7xD4, C7xD4, C7xD4, C7xQ8, C22xC14, C7xM4(2), C7xD8, C7xSD16, D4xC14, C7xC4oD4, C7xC8:C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2xD4, C2xC14, C8:C22, C7xD4, C22xC14, D4xC14, C7xC8:C22
►On 56 points
(1 26 24 51 39 43 16)(2 27 17 52 40 44 9)(3 28 18 53 33 45 10)(4 29 19 54 34 46 11)(5 30 20 55 35 47 12)(6 31 21 56 36 48 13)(7 32 22 49 37 41 14)(8 25 23 50 38 42 15)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)
G:=sub<Sym(56)| (1,26,24,51,39,43,16)(2,27,17,52,40,44,9)(3,28,18,53,33,45,10)(4,29,19,54,34,46,11)(5,30,20,55,35,47,12)(6,31,21,56,36,48,13)(7,32,22,49,37,41,14)(8,25,23,50,38,42,15), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)>;
G:=Group( (1,26,24,51,39,43,16)(2,27,17,52,40,44,9)(3,28,18,53,33,45,10)(4,29,19,54,34,46,11)(5,30,20,55,35,47,12)(6,31,21,56,36,48,13)(7,32,22,49,37,41,14)(8,25,23,50,38,42,15), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56) );
G=PermutationGroup([[(1,26,24,51,39,43,16),(2,27,17,52,40,44,9),(3,28,18,53,33,45,10),(4,29,19,54,34,46,11),(5,30,20,55,35,47,12),(6,31,21,56,36,48,13),(7,32,22,49,37,41,14),(8,25,23,50,38,42,15)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,31),(27,29),(28,32),(33,37),(34,40),(36,38),(41,45),(42,48),(44,46),(49,53),(50,56),(52,54)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56)]])
C7xC8:C22 is a maximal subgroup of
D28:18D4 M4(2).D14 M4(2).13D14 D28.38D4 SD16:D14 D8:5D14 D8:6D14
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 1 | ··· | 1 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7xD4 | C7xD4 | C8:C22 | C7xC8:C22 |
| kernel | C7xC8:C22 | C7xM4(2) | C7xD8 | C7xSD16 | D4xC14 | C7xC4oD4 | C8:C22 | M4(2) | D8 | SD16 | C2xD4 | C4oD4 | C28 | C2xC14 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 1 | 1 | 6 | 6 | 1 | 6 |
Matrix representation of C7xC8:C22 ►in GL4(F113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 18 | 40 | 63 | 106 |
| 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 111 | 112 | 62 | 95 |
| 1 | 1 | 91 | 91 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 112 | 0 |
| 1 | 0 | 51 | 18 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[18,0,0,111,40,0,1,112,63,0,0,62,106,112,0,95],[1,0,0,0,1,112,0,0,91,0,0,112,91,0,112,0],[1,0,0,0,0,1,0,0,51,0,112,0,18,0,0,112] >;
C7xC8:C22 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes C_2^2
% in TeX
G:=Group("C7xC8:C2^2"); // GroupNames label
G:=SmallGroup(224,171);
// by ID
G=gap.SmallGroup(224,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,2090,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations