direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22.53C24, C22.68C25, C23.29C24, C42.565C23, C24.499C23, C22.1162+ 1+4, (C2×C4).66C24, (C4×Q8)⋊97C22, (C4×D4)⋊109C22, C4⋊C4.477C23, (C2×D4).460C23, C4.4D4⋊76C22, C22⋊C4.90C23, (C2×Q8).436C23, C4⋊1D4.181C22, (C2×C42).936C22, (C23×C4).188C22, (C22×C4).618C23, C2.23(C2×2+ 1+4), (C22×D4).595C22, C22.D4⋊46C22, (C22×Q8).496C22, (C2×C4×D4)⋊88C2, (C2×C4×Q8)⋊56C2, C4.176(C2×C4○D4), (C2×C4⋊1D4).27C2, (C2×C4.4D4)⋊53C2, C2.39(C22×C4○D4), (C2×C4).909(C4○D4), (C2×C4⋊C4).960C22, C22.163(C2×C4○D4), (C2×C22.D4)⋊58C2, (C2×C22⋊C4).545C22, SmallGroup(128,2211)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.53C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=c, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, 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, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, C23×C4, C22×D4, C22×Q8, C2×C4×D4, C2×C4×Q8, C2×C22.D4, C2×C4.4D4, C2×C4⋊1D4, C22.53C24, C2×C22.53C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22.53C24, C22×C4○D4, C2×2+ 1+4, C2×C22.53C24
►On 64 points
Generators in S64
(1 19)(2 20)(3 17)(4 18)(5 60)(6 57)(7 58)(8 59)(9 52)(10 49)(11 50)(12 51)(13 30)(14 31)(15 32)(16 29)(21 39)(22 40)(23 37)(24 38)(25 55)(26 56)(27 53)(28 54)(33 43)(34 44)(35 41)(36 42)(45 61)(46 62)(47 63)(48 64)
(1 53)(2 54)(3 55)(4 56)(5 22)(6 23)(7 24)(8 21)(9 30)(10 31)(11 32)(12 29)(13 52)(14 49)(15 50)(16 51)(17 25)(18 26)(19 27)(20 28)(33 64)(34 61)(35 62)(36 63)(37 57)(38 58)(39 59)(40 60)(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 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 32 55 9)(2 12 56 31)(3 30 53 11)(4 10 54 29)(5 36 24 61)(6 64 21 35)(7 34 22 63)(8 62 23 33)(13 27 50 17)(14 20 51 26)(15 25 52 19)(16 18 49 28)(37 43 59 46)(38 45 60 42)(39 41 57 48)(40 47 58 44)
(1 47 3 45)(2 46 4 48)(5 15 7 13)(6 14 8 16)(9 40 11 38)(10 39 12 37)(17 61 19 63)(18 64 20 62)(21 51 23 49)(22 50 24 52)(25 34 27 36)(26 33 28 35)(29 57 31 59)(30 60 32 58)(41 56 43 54)(42 55 44 53)
(1 29 53 12)(2 30 54 9)(3 31 55 10)(4 32 56 11)(5 35 22 62)(6 36 23 63)(7 33 24 64)(8 34 21 61)(13 28 52 20)(14 25 49 17)(15 26 50 18)(16 27 51 19)(37 47 57 42)(38 48 58 43)(39 45 59 44)(40 46 60 41)
G:=sub<Sym(64)| (1,19)(2,20)(3,17)(4,18)(5,60)(6,57)(7,58)(8,59)(9,52)(10,49)(11,50)(12,51)(13,30)(14,31)(15,32)(16,29)(21,39)(22,40)(23,37)(24,38)(25,55)(26,56)(27,53)(28,54)(33,43)(34,44)(35,41)(36,42)(45,61)(46,62)(47,63)(48,64), (1,53)(2,54)(3,55)(4,56)(5,22)(6,23)(7,24)(8,21)(9,30)(10,31)(11,32)(12,29)(13,52)(14,49)(15,50)(16,51)(17,25)(18,26)(19,27)(20,28)(33,64)(34,61)(35,62)(36,63)(37,57)(38,58)(39,59)(40,60)(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), (1,2,3,4)(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), (1,32,55,9)(2,12,56,31)(3,30,53,11)(4,10,54,29)(5,36,24,61)(6,64,21,35)(7,34,22,63)(8,62,23,33)(13,27,50,17)(14,20,51,26)(15,25,52,19)(16,18,49,28)(37,43,59,46)(38,45,60,42)(39,41,57,48)(40,47,58,44), (1,47,3,45)(2,46,4,48)(5,15,7,13)(6,14,8,16)(9,40,11,38)(10,39,12,37)(17,61,19,63)(18,64,20,62)(21,51,23,49)(22,50,24,52)(25,34,27,36)(26,33,28,35)(29,57,31,59)(30,60,32,58)(41,56,43,54)(42,55,44,53), (1,29,53,12)(2,30,54,9)(3,31,55,10)(4,32,56,11)(5,35,22,62)(6,36,23,63)(7,33,24,64)(8,34,21,61)(13,28,52,20)(14,25,49,17)(15,26,50,18)(16,27,51,19)(37,47,57,42)(38,48,58,43)(39,45,59,44)(40,46,60,41)>;
G:=Group( (1,19)(2,20)(3,17)(4,18)(5,60)(6,57)(7,58)(8,59)(9,52)(10,49)(11,50)(12,51)(13,30)(14,31)(15,32)(16,29)(21,39)(22,40)(23,37)(24,38)(25,55)(26,56)(27,53)(28,54)(33,43)(34,44)(35,41)(36,42)(45,61)(46,62)(47,63)(48,64), (1,53)(2,54)(3,55)(4,56)(5,22)(6,23)(7,24)(8,21)(9,30)(10,31)(11,32)(12,29)(13,52)(14,49)(15,50)(16,51)(17,25)(18,26)(19,27)(20,28)(33,64)(34,61)(35,62)(36,63)(37,57)(38,58)(39,59)(40,60)(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), (1,2,3,4)(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), (1,32,55,9)(2,12,56,31)(3,30,53,11)(4,10,54,29)(5,36,24,61)(6,64,21,35)(7,34,22,63)(8,62,23,33)(13,27,50,17)(14,20,51,26)(15,25,52,19)(16,18,49,28)(37,43,59,46)(38,45,60,42)(39,41,57,48)(40,47,58,44), (1,47,3,45)(2,46,4,48)(5,15,7,13)(6,14,8,16)(9,40,11,38)(10,39,12,37)(17,61,19,63)(18,64,20,62)(21,51,23,49)(22,50,24,52)(25,34,27,36)(26,33,28,35)(29,57,31,59)(30,60,32,58)(41,56,43,54)(42,55,44,53), (1,29,53,12)(2,30,54,9)(3,31,55,10)(4,32,56,11)(5,35,22,62)(6,36,23,63)(7,33,24,64)(8,34,21,61)(13,28,52,20)(14,25,49,17)(15,26,50,18)(16,27,51,19)(37,47,57,42)(38,48,58,43)(39,45,59,44)(40,46,60,41) );
G=PermutationGroup([[(1,19),(2,20),(3,17),(4,18),(5,60),(6,57),(7,58),(8,59),(9,52),(10,49),(11,50),(12,51),(13,30),(14,31),(15,32),(16,29),(21,39),(22,40),(23,37),(24,38),(25,55),(26,56),(27,53),(28,54),(33,43),(34,44),(35,41),(36,42),(45,61),(46,62),(47,63),(48,64)], [(1,53),(2,54),(3,55),(4,56),(5,22),(6,23),(7,24),(8,21),(9,30),(10,31),(11,32),(12,29),(13,52),(14,49),(15,50),(16,51),(17,25),(18,26),(19,27),(20,28),(33,64),(34,61),(35,62),(36,63),(37,57),(38,58),(39,59),(40,60),(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)], [(1,2,3,4),(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)], [(1,32,55,9),(2,12,56,31),(3,30,53,11),(4,10,54,29),(5,36,24,61),(6,64,21,35),(7,34,22,63),(8,62,23,33),(13,27,50,17),(14,20,51,26),(15,25,52,19),(16,18,49,28),(37,43,59,46),(38,45,60,42),(39,41,57,48),(40,47,58,44)], [(1,47,3,45),(2,46,4,48),(5,15,7,13),(6,14,8,16),(9,40,11,38),(10,39,12,37),(17,61,19,63),(18,64,20,62),(21,51,23,49),(22,50,24,52),(25,34,27,36),(26,33,28,35),(29,57,31,59),(30,60,32,58),(41,56,43,54),(42,55,44,53)], [(1,29,53,12),(2,30,54,9),(3,31,55,10),(4,32,56,11),(5,35,22,62),(6,36,23,63),(7,33,24,64),(8,34,21,61),(13,28,52,20),(14,25,49,17),(15,26,50,18),(16,27,51,19),(37,47,57,42),(38,48,58,43),(39,45,59,44),(40,46,60,41)]])
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×C22.53C24 | C2×C4×D4 | C2×C4×Q8 | C2×C22.D4 | C2×C4.4D4 | C2×C4⋊1D4 | C22.53C24 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 4 | 4 | 1 | 16 | 16 | 2 |
Matrix representation of C2×C22.53C24 ►in GL5(𝔽5)
| 4 | 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 |
| 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 | 0 | 3 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[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],[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],[1,0,0,0,0,0,0,2,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,3],[1,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,1] >;
C2×C22.53C24 in GAP, Magma, Sage, TeX
C_2\times C_2^2._{53}C_2^4 % in TeX
G:=Group("C2xC2^2.53C2^4"); // GroupNames label
G:=SmallGroup(128,2211);
// by ID
G=gap.SmallGroup(128,2211);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,680,1430,352,570,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=c,e^2=c*b=b*c,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,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