direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22.49C24, C22.62C25, C23.28C24, C24.496C23, C42.560C23, C22.1142+ 1+4, C4⋊Q8⋊85C22, (C2×C4).61C24, (C4×D4)⋊108C22, C4⋊D4⋊75C22, C4⋊C4.518C23, (C2×D4).457C23, C4.4D4⋊74C22, C22⋊C4.86C23, (C2×Q8).280C23, C42⋊C2⋊97C22, (C23×C4).600C22, (C2×C42).931C22, C2.21(C2×2+ 1+4), (C22×C4).1198C23, (C22×D4).423C22, (C22×Q8).355C22, (C2×C4×D4)⋊86C2, (C2×C4⋊Q8)⋊53C2, (C2×C4⋊D4)⋊64C2, C4.135(C2×C4○D4), (C2×C4.4D4)⋊52C2, C2.34(C22×C4○D4), (C2×C42⋊C2)⋊62C2, (C2×C4).851(C4○D4), (C2×C4⋊C4).983C22, C22.160(C2×C4○D4), (C2×C22⋊C4).542C22, SmallGroup(128,2205)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.49C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 956 in 620 conjugacy classes, 404 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4, C2×C4.4D4, C2×C4⋊Q8, C22.49C24, C2×C22.49C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22.49C24, C22×C4○D4, C2×2+ 1+4, C2×C22.49C24
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 37)(20 38)(21 46)(22 47)(23 48)(24 45)(33 44)(34 41)(35 42)(36 43)(49 60)(50 57)(51 58)(52 59)(53 64)(54 61)(55 62)(56 63)
(1 11)(2 12)(3 9)(4 10)(5 37)(6 38)(7 39)(8 40)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(21 54)(22 55)(23 56)(24 53)(33 60)(34 57)(35 58)(36 59)(41 50)(42 51)(43 52)(44 49)(45 64)(46 61)(47 62)(48 63)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 59)(2 33)(3 57)(4 35)(5 63)(6 45)(7 61)(8 47)(9 34)(10 58)(11 36)(12 60)(13 41)(14 51)(15 43)(16 49)(17 21)(18 55)(19 23)(20 53)(22 32)(24 30)(25 52)(26 44)(27 50)(28 42)(29 56)(31 54)(37 48)(38 64)(39 46)(40 62)
(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)
(1 5 11 37)(2 6 12 38)(3 7 9 39)(4 8 10 40)(13 17 27 31)(14 18 28 32)(15 19 25 29)(16 20 26 30)(21 52 54 43)(22 49 55 44)(23 50 56 41)(24 51 53 42)(33 47 60 62)(34 48 57 63)(35 45 58 64)(36 46 59 61)
(1 31 11 17)(2 18 12 32)(3 29 9 19)(4 20 10 30)(5 13 37 27)(6 28 38 14)(7 15 39 25)(8 26 40 16)(21 59 54 36)(22 33 55 60)(23 57 56 34)(24 35 53 58)(41 48 50 63)(42 64 51 45)(43 46 52 61)(44 62 49 47)
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,37)(20,38)(21,46)(22,47)(23,48)(24,45)(33,44)(34,41)(35,42)(36,43)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (1,11)(2,12)(3,9)(4,10)(5,37)(6,38)(7,39)(8,40)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(21,54)(22,55)(23,56)(24,53)(33,60)(34,57)(35,58)(36,59)(41,50)(42,51)(43,52)(44,49)(45,64)(46,61)(47,62)(48,63), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,59)(2,33)(3,57)(4,35)(5,63)(6,45)(7,61)(8,47)(9,34)(10,58)(11,36)(12,60)(13,41)(14,51)(15,43)(16,49)(17,21)(18,55)(19,23)(20,53)(22,32)(24,30)(25,52)(26,44)(27,50)(28,42)(29,56)(31,54)(37,48)(38,64)(39,46)(40,62), (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), (1,5,11,37)(2,6,12,38)(3,7,9,39)(4,8,10,40)(13,17,27,31)(14,18,28,32)(15,19,25,29)(16,20,26,30)(21,52,54,43)(22,49,55,44)(23,50,56,41)(24,51,53,42)(33,47,60,62)(34,48,57,63)(35,45,58,64)(36,46,59,61), (1,31,11,17)(2,18,12,32)(3,29,9,19)(4,20,10,30)(5,13,37,27)(6,28,38,14)(7,15,39,25)(8,26,40,16)(21,59,54,36)(22,33,55,60)(23,57,56,34)(24,35,53,58)(41,48,50,63)(42,64,51,45)(43,46,52,61)(44,62,49,47)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,37)(20,38)(21,46)(22,47)(23,48)(24,45)(33,44)(34,41)(35,42)(36,43)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (1,11)(2,12)(3,9)(4,10)(5,37)(6,38)(7,39)(8,40)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(21,54)(22,55)(23,56)(24,53)(33,60)(34,57)(35,58)(36,59)(41,50)(42,51)(43,52)(44,49)(45,64)(46,61)(47,62)(48,63), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,59)(2,33)(3,57)(4,35)(5,63)(6,45)(7,61)(8,47)(9,34)(10,58)(11,36)(12,60)(13,41)(14,51)(15,43)(16,49)(17,21)(18,55)(19,23)(20,53)(22,32)(24,30)(25,52)(26,44)(27,50)(28,42)(29,56)(31,54)(37,48)(38,64)(39,46)(40,62), (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), (1,5,11,37)(2,6,12,38)(3,7,9,39)(4,8,10,40)(13,17,27,31)(14,18,28,32)(15,19,25,29)(16,20,26,30)(21,52,54,43)(22,49,55,44)(23,50,56,41)(24,51,53,42)(33,47,60,62)(34,48,57,63)(35,45,58,64)(36,46,59,61), (1,31,11,17)(2,18,12,32)(3,29,9,19)(4,20,10,30)(5,13,37,27)(6,28,38,14)(7,15,39,25)(8,26,40,16)(21,59,54,36)(22,33,55,60)(23,57,56,34)(24,35,53,58)(41,48,50,63)(42,64,51,45)(43,46,52,61)(44,62,49,47) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,13),(10,14),(11,15),(12,16),(17,39),(18,40),(19,37),(20,38),(21,46),(22,47),(23,48),(24,45),(33,44),(34,41),(35,42),(36,43),(49,60),(50,57),(51,58),(52,59),(53,64),(54,61),(55,62),(56,63)], [(1,11),(2,12),(3,9),(4,10),(5,37),(6,38),(7,39),(8,40),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(21,54),(22,55),(23,56),(24,53),(33,60),(34,57),(35,58),(36,59),(41,50),(42,51),(43,52),(44,49),(45,64),(46,61),(47,62),(48,63)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,59),(2,33),(3,57),(4,35),(5,63),(6,45),(7,61),(8,47),(9,34),(10,58),(11,36),(12,60),(13,41),(14,51),(15,43),(16,49),(17,21),(18,55),(19,23),(20,53),(22,32),(24,30),(25,52),(26,44),(27,50),(28,42),(29,56),(31,54),(37,48),(38,64),(39,46),(40,62)], [(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)], [(1,5,11,37),(2,6,12,38),(3,7,9,39),(4,8,10,40),(13,17,27,31),(14,18,28,32),(15,19,25,29),(16,20,26,30),(21,52,54,43),(22,49,55,44),(23,50,56,41),(24,51,53,42),(33,47,60,62),(34,48,57,63),(35,45,58,64),(36,46,59,61)], [(1,31,11,17),(2,18,12,32),(3,29,9,19),(4,20,10,30),(5,13,37,27),(6,28,38,14),(7,15,39,25),(8,26,40,16),(21,59,54,36),(22,33,55,60),(23,57,56,34),(24,35,53,58),(41,48,50,63),(42,64,51,45),(43,46,52,61),(44,62,49,47)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AH |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ 1+4 |
| kernel | C2×C22.49C24 | C2×C42⋊C2 | C2×C4×D4 | C2×C4⋊D4 | C2×C4.4D4 | C2×C4⋊Q8 | C22.49C24 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 4 | 4 | 1 | 16 | 16 | 2 |
Matrix representation of C2×C22.49C24 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,3,4],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4,4,0,0,0,0,1],[4,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4] >;
C2×C22.49C24 in GAP, Magma, Sage, TeX
C_2\times C_2^2._{49}C_2^4 % in TeX
G:=Group("C2xC2^2.49C2^4"); // GroupNames label
G:=SmallGroup(128,2205);
// by ID
G=gap.SmallGroup(128,2205);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations