Copied to
clipboard

G = C4218D4order 128 = 27

12nd semidirect product of C42 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C4218D4, C24.320C23, C23.437C24, C22.2262+ 1+4, C22.1742- 1+4, C2.28D42, (C2×D4)⋊34D4, C45(C4⋊D4), C23.47(C2×D4), C429C426C2, C2.49(D46D4), (C22×C4).94C23, C23.7Q865C2, C23.10D439C2, (C2×C42).543C22, (C23×C4).390C22, C22.288(C22×D4), (C22×D4).161C22, C2.C42.180C22, C2.12(C22.31C24), C2.24(C22.49C24), (C2×C4×D4)⋊43C2, (C2×C4⋊D4)⋊15C2, (C2×C4).351(C2×D4), C2.32(C2×C4⋊D4), (C2×C4).819(C4○D4), (C2×C4⋊C4).297C22, C22.314(C2×C4○D4), (C2×C22⋊C4).173C22, SmallGroup(128,1269)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4218D4
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C4218D4
C1C23 — C4218D4
C1C23 — C4218D4
C1C23 — C4218D4

Generators and relations for C4218D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 804 in 394 conjugacy classes, 124 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×10], C22 [×3], C22 [×4], C22 [×40], C2×C4 [×14], C2×C4 [×42], D4 [×32], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×14], C22×C4, C22×C4 [×10], C22×C4 [×24], C2×D4 [×8], C2×D4 [×28], C24 [×4], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×8], C4×D4 [×8], C4⋊D4 [×16], C23×C4 [×4], C22×D4 [×6], C23.7Q8 [×4], C429C4, C23.10D4 [×4], C2×C4×D4 [×2], C2×C4⋊D4 [×4], C4218D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C4⋊D4 [×8], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C2×C4⋊D4 [×2], C22.31C24, D42, D46D4 [×2], C22.49C24, C4218D4

Smallest permutation representation of C4218D4
On 64 points
Generators in S64
(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 15 11 20)(2 16 12 17)(3 13 9 18)(4 14 10 19)(5 29 28 63)(6 30 25 64)(7 31 26 61)(8 32 27 62)(21 44 45 52)(22 41 46 49)(23 42 47 50)(24 43 48 51)(33 38 55 59)(34 39 56 60)(35 40 53 57)(36 37 54 58)
(1 32 44 39)(2 31 41 38)(3 30 42 37)(4 29 43 40)(5 48 35 14)(6 47 36 13)(7 46 33 16)(8 45 34 15)(9 64 50 58)(10 63 51 57)(11 62 52 60)(12 61 49 59)(17 26 22 55)(18 25 23 54)(19 28 24 53)(20 27 21 56)
(1 33)(2 56)(3 35)(4 54)(5 42)(6 51)(7 44)(8 49)(9 53)(10 36)(11 55)(12 34)(13 57)(14 37)(15 59)(16 39)(17 60)(18 40)(19 58)(20 38)(21 31)(22 62)(23 29)(24 64)(25 43)(26 52)(27 41)(28 50)(30 48)(32 46)(45 61)(47 63)

G:=sub<Sym(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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,32,44,39)(2,31,41,38)(3,30,42,37)(4,29,43,40)(5,48,35,14)(6,47,36,13)(7,46,33,16)(8,45,34,15)(9,64,50,58)(10,63,51,57)(11,62,52,60)(12,61,49,59)(17,26,22,55)(18,25,23,54)(19,28,24,53)(20,27,21,56), (1,33)(2,56)(3,35)(4,54)(5,42)(6,51)(7,44)(8,49)(9,53)(10,36)(11,55)(12,34)(13,57)(14,37)(15,59)(16,39)(17,60)(18,40)(19,58)(20,38)(21,31)(22,62)(23,29)(24,64)(25,43)(26,52)(27,41)(28,50)(30,48)(32,46)(45,61)(47,63)>;

G:=Group( (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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,32,44,39)(2,31,41,38)(3,30,42,37)(4,29,43,40)(5,48,35,14)(6,47,36,13)(7,46,33,16)(8,45,34,15)(9,64,50,58)(10,63,51,57)(11,62,52,60)(12,61,49,59)(17,26,22,55)(18,25,23,54)(19,28,24,53)(20,27,21,56), (1,33)(2,56)(3,35)(4,54)(5,42)(6,51)(7,44)(8,49)(9,53)(10,36)(11,55)(12,34)(13,57)(14,37)(15,59)(16,39)(17,60)(18,40)(19,58)(20,38)(21,31)(22,62)(23,29)(24,64)(25,43)(26,52)(27,41)(28,50)(30,48)(32,46)(45,61)(47,63) );

G=PermutationGroup([(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)], [(1,15,11,20),(2,16,12,17),(3,13,9,18),(4,14,10,19),(5,29,28,63),(6,30,25,64),(7,31,26,61),(8,32,27,62),(21,44,45,52),(22,41,46,49),(23,42,47,50),(24,43,48,51),(33,38,55,59),(34,39,56,60),(35,40,53,57),(36,37,54,58)], [(1,32,44,39),(2,31,41,38),(3,30,42,37),(4,29,43,40),(5,48,35,14),(6,47,36,13),(7,46,33,16),(8,45,34,15),(9,64,50,58),(10,63,51,57),(11,62,52,60),(12,61,49,59),(17,26,22,55),(18,25,23,54),(19,28,24,53),(20,27,21,56)], [(1,33),(2,56),(3,35),(4,54),(5,42),(6,51),(7,44),(8,49),(9,53),(10,36),(11,55),(12,34),(13,57),(14,37),(15,59),(16,39),(17,60),(18,40),(19,58),(20,38),(21,31),(22,62),(23,29),(24,64),(25,43),(26,52),(27,41),(28,50),(30,48),(32,46),(45,61),(47,63)])

38 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4R4S4T4U4V
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim11111122244
type+++++++++-
imageC1C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC4218D4C23.7Q8C429C4C23.10D4C2×C4×D4C2×C4⋊D4C42C2×D4C2×C4C22C22
# reps14142448811

Matrix representation of C4218D4 in GL6(𝔽5)

100000
010000
000100
004000
000020
000023
,
100000
010000
004000
000400
000020
000023
,
420000
410000
002000
000300
000013
000014
,
420000
010000
000300
002000
000042
000001

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;

C4218D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{18}D_4
% in TeX

G:=Group("C4^2:18D4");
// GroupNames label

G:=SmallGroup(128,1269);
// by ID

G=gap.SmallGroup(128,1269);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,80]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽