direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22.46C24, C22.59C25, C23.26C24, C24.616C23, C42.557C23, C22.822- 1+4, (C2×C4).59C24, (C4×Q8)⋊94C22, C4⋊C4.470C23, C22⋊Q8⋊88C22, (C4×D4).351C22, (C2×D4).455C23, C22⋊C4.14C23, (C2×Q8).432C23, C42.C2⋊47C22, C42⋊2C2⋊29C22, C42⋊C2⋊95C22, C23.320(C4○D4), (C23×C4).597C22, (C2×C42).928C22, C2.14(C2×2- 1+4), (C22×C4).1195C23, (C22×D4).591C22, (C22×Q8).492C22, C22.D4.26C22, (C2×C4×Q8)⋊52C2, (C2×C4×D4).87C2, (C22×C4⋊C4)⋊44C2, C4.132(C2×C4○D4), (C2×C22⋊Q8)⋊72C2, (C2×C4⋊C4)⋊136C22, (C2×C42.C2)⋊42C2, C2.31(C22×C4○D4), C22.25(C2×C4○D4), (C2×C42⋊2C2)⋊35C2, (C2×C42⋊C2)⋊60C2, (C2×C4).848(C4○D4), (C2×C22⋊C4).378C22, (C2×C22.D4).33C2, SmallGroup(128,2202)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.46C24
G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 732 in 562 conjugacy classes, 404 normal (32 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×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C23×C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, C2×C42⋊C2, C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8, C2×C22.D4, C2×C42.C2, C2×C42.C2, C2×C42⋊2C2, C22.46C24, C2×C22.46C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22.46C24, C22×C4○D4, C2×2- 1+4, C2×C22.46C24
►On 64 points
Generators in S64
(1 59)(2 60)(3 57)(4 58)(5 27)(6 28)(7 25)(8 26)(9 21)(10 22)(11 23)(12 24)(13 19)(14 20)(15 17)(16 18)(29 64)(30 61)(31 62)(32 63)(33 56)(34 53)(35 54)(36 55)(37 52)(38 49)(39 50)(40 51)(41 46)(42 47)(43 48)(44 45)
(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 62)(2 63)(3 64)(4 61)(5 34)(6 35)(7 36)(8 33)(9 40)(10 37)(11 38)(12 39)(13 42)(14 43)(15 44)(16 41)(17 45)(18 46)(19 47)(20 48)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 57)(30 58)(31 59)(32 60)
(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 14 62 43)(2 13 63 42)(3 16 64 41)(4 15 61 44)(5 37 34 10)(6 40 35 9)(7 39 36 12)(8 38 33 11)(17 30 45 58)(18 29 46 57)(19 32 47 60)(20 31 48 59)(21 28 51 54)(22 27 52 53)(23 26 49 56)(24 25 50 55)
(1 49 3 51)(2 24 4 22)(5 19 7 17)(6 48 8 46)(9 31 11 29)(10 60 12 58)(13 25 15 27)(14 56 16 54)(18 35 20 33)(21 62 23 64)(26 41 28 43)(30 37 32 39)(34 47 36 45)(38 57 40 59)(42 55 44 53)(50 61 52 63)
(1 25)(2 26)(3 27)(4 28)(5 57)(6 58)(7 59)(8 60)(9 47)(10 48)(11 45)(12 46)(13 51)(14 52)(15 49)(16 50)(17 38)(18 39)(19 40)(20 37)(21 42)(22 43)(23 44)(24 41)(29 34)(30 35)(31 36)(32 33)(53 64)(54 61)(55 62)(56 63)
G:=sub<Sym(64)| (1,59)(2,60)(3,57)(4,58)(5,27)(6,28)(7,25)(8,26)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18)(29,64)(30,61)(31,62)(32,63)(33,56)(34,53)(35,54)(36,55)(37,52)(38,49)(39,50)(40,51)(41,46)(42,47)(43,48)(44,45), (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), (1,62)(2,63)(3,64)(4,61)(5,34)(6,35)(7,36)(8,33)(9,40)(10,37)(11,38)(12,39)(13,42)(14,43)(15,44)(16,41)(17,45)(18,46)(19,47)(20,48)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,57)(30,58)(31,59)(32,60), (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), (1,14,62,43)(2,13,63,42)(3,16,64,41)(4,15,61,44)(5,37,34,10)(6,40,35,9)(7,39,36,12)(8,38,33,11)(17,30,45,58)(18,29,46,57)(19,32,47,60)(20,31,48,59)(21,28,51,54)(22,27,52,53)(23,26,49,56)(24,25,50,55), (1,49,3,51)(2,24,4,22)(5,19,7,17)(6,48,8,46)(9,31,11,29)(10,60,12,58)(13,25,15,27)(14,56,16,54)(18,35,20,33)(21,62,23,64)(26,41,28,43)(30,37,32,39)(34,47,36,45)(38,57,40,59)(42,55,44,53)(50,61,52,63), (1,25)(2,26)(3,27)(4,28)(5,57)(6,58)(7,59)(8,60)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,38)(18,39)(19,40)(20,37)(21,42)(22,43)(23,44)(24,41)(29,34)(30,35)(31,36)(32,33)(53,64)(54,61)(55,62)(56,63)>;
G:=Group( (1,59)(2,60)(3,57)(4,58)(5,27)(6,28)(7,25)(8,26)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18)(29,64)(30,61)(31,62)(32,63)(33,56)(34,53)(35,54)(36,55)(37,52)(38,49)(39,50)(40,51)(41,46)(42,47)(43,48)(44,45), (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), (1,62)(2,63)(3,64)(4,61)(5,34)(6,35)(7,36)(8,33)(9,40)(10,37)(11,38)(12,39)(13,42)(14,43)(15,44)(16,41)(17,45)(18,46)(19,47)(20,48)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,57)(30,58)(31,59)(32,60), (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), (1,14,62,43)(2,13,63,42)(3,16,64,41)(4,15,61,44)(5,37,34,10)(6,40,35,9)(7,39,36,12)(8,38,33,11)(17,30,45,58)(18,29,46,57)(19,32,47,60)(20,31,48,59)(21,28,51,54)(22,27,52,53)(23,26,49,56)(24,25,50,55), (1,49,3,51)(2,24,4,22)(5,19,7,17)(6,48,8,46)(9,31,11,29)(10,60,12,58)(13,25,15,27)(14,56,16,54)(18,35,20,33)(21,62,23,64)(26,41,28,43)(30,37,32,39)(34,47,36,45)(38,57,40,59)(42,55,44,53)(50,61,52,63), (1,25)(2,26)(3,27)(4,28)(5,57)(6,58)(7,59)(8,60)(9,47)(10,48)(11,45)(12,46)(13,51)(14,52)(15,49)(16,50)(17,38)(18,39)(19,40)(20,37)(21,42)(22,43)(23,44)(24,41)(29,34)(30,35)(31,36)(32,33)(53,64)(54,61)(55,62)(56,63) );
G=PermutationGroup([[(1,59),(2,60),(3,57),(4,58),(5,27),(6,28),(7,25),(8,26),(9,21),(10,22),(11,23),(12,24),(13,19),(14,20),(15,17),(16,18),(29,64),(30,61),(31,62),(32,63),(33,56),(34,53),(35,54),(36,55),(37,52),(38,49),(39,50),(40,51),(41,46),(42,47),(43,48),(44,45)], [(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)], [(1,62),(2,63),(3,64),(4,61),(5,34),(6,35),(7,36),(8,33),(9,40),(10,37),(11,38),(12,39),(13,42),(14,43),(15,44),(16,41),(17,45),(18,46),(19,47),(20,48),(21,51),(22,52),(23,49),(24,50),(25,55),(26,56),(27,53),(28,54),(29,57),(30,58),(31,59),(32,60)], [(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)], [(1,14,62,43),(2,13,63,42),(3,16,64,41),(4,15,61,44),(5,37,34,10),(6,40,35,9),(7,39,36,12),(8,38,33,11),(17,30,45,58),(18,29,46,57),(19,32,47,60),(20,31,48,59),(21,28,51,54),(22,27,52,53),(23,26,49,56),(24,25,50,55)], [(1,49,3,51),(2,24,4,22),(5,19,7,17),(6,48,8,46),(9,31,11,29),(10,60,12,58),(13,25,15,27),(14,56,16,54),(18,35,20,33),(21,62,23,64),(26,41,28,43),(30,37,32,39),(34,47,36,45),(38,57,40,59),(42,55,44,53),(50,61,52,63)], [(1,25),(2,26),(3,27),(4,28),(5,57),(6,58),(7,59),(8,60),(9,47),(10,48),(11,45),(12,46),(13,51),(14,52),(15,49),(16,50),(17,38),(18,39),(19,40),(20,37),(21,42),(22,43),(23,44),(24,41),(29,34),(30,35),(31,36),(32,33),(53,64),(54,61),(55,62),(56,63)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4T | 4U | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2- 1+4 |
| kernel | C2×C22.46C24 | C22×C4⋊C4 | C2×C42⋊C2 | C2×C4×D4 | C2×C4×Q8 | C2×C22⋊Q8 | C2×C22.D4 | C2×C42.C2 | C2×C42⋊2C2 | C22.46C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 3 | 1 | 1 | 2 | 2 | 3 | 2 | 16 | 8 | 8 | 2 |
Matrix representation of C2×C22.46C24 ►in GL5(𝔽5)
| 4 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 2 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,3,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0],[4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,4,0] >;
C2×C22.46C24 in GAP, Magma, Sage, TeX
C_2\times C_2^2._{46}C_2^4 % in TeX
G:=Group("C2xC2^2.46C2^4"); // GroupNames label
G:=SmallGroup(128,2202);
// by ID
G=gap.SmallGroup(128,2202);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,184,570]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=f^2=b,e^2=c,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=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