Copied to
clipboard

G = C4232D4order 128 = 27

26th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4232D4, C24.54C23, C23.562C24, C22.2512- 1+4, C22.3362+ 1+4, C23⋊Q835C2, C232D4.20C2, C23.Q846C2, C23.10D467C2, (C22×C4).167C23, (C2×C42).626C22, C22.374(C22×D4), C24.3C2270C2, (C22×D4).210C22, (C22×Q8).168C22, C24.C22112C2, C23.78C2333C2, C2.50(C22.29C24), C2.53(C22.32C24), C2.C42.276C22, C2.51(C22.26C24), C2.34(C22.31C24), C2.63(C22.36C24), (C4×C4⋊C4)⋊115C2, (C2×C4).407(C2×D4), (C2×C4.4D4)⋊23C2, (C2×C422C2)⋊16C2, (C2×C4).182(C4○D4), (C2×C4⋊C4).896C22, C22.429(C2×C4○D4), (C2×C22⋊C4).240C22, SmallGroup(128,1394)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4232D4
C1C2C22C23C22×C4C2×C22⋊C4C24.3C22 — C4232D4
C1C23 — C4232D4
C1C23 — C4232D4
C1C23 — C4232D4

Generators and relations for C4232D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 580 in 267 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×3], C4 [×17], C22 [×3], C22 [×4], C22 [×21], C2×C4 [×10], C2×C4 [×31], D4 [×12], Q8 [×4], C23, C23 [×21], C42 [×4], C42 [×2], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×6], C2×D4 [×12], C2×Q8 [×4], C24, C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C4.4D4 [×4], C422C2 [×4], C22×D4, C22×D4 [×2], C22×Q8, C4×C4⋊C4, C24.C22 [×2], C24.3C22 [×2], C232D4, C23⋊Q8, C23.10D4, C23.10D4 [×2], C23.78C23, C23.Q8 [×2], C2×C4.4D4, C2×C422C2, C4232D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C22.26C24, C22.29C24, C22.31C24, C22.32C24 [×2], C22.36C24 [×2], C4232D4

Smallest permutation representation of C4232D4
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 13 59 28)(2 14 60 25)(3 15 57 26)(4 16 58 27)(5 36 45 22)(6 33 46 23)(7 34 47 24)(8 35 48 21)(9 55 32 44)(10 56 29 41)(11 53 30 42)(12 54 31 43)(17 63 39 52)(18 64 40 49)(19 61 37 50)(20 62 38 51)
(1 61 53 48)(2 64 54 47)(3 63 55 46)(4 62 56 45)(5 58 51 41)(6 57 52 44)(7 60 49 43)(8 59 50 42)(9 33 26 17)(10 36 27 20)(11 35 28 19)(12 34 25 18)(13 37 30 21)(14 40 31 24)(15 39 32 23)(16 38 29 22)
(1 48)(2 7)(3 46)(4 5)(6 57)(8 59)(9 19)(10 40)(11 17)(12 38)(13 23)(14 36)(15 21)(16 34)(18 29)(20 31)(22 25)(24 27)(26 35)(28 33)(30 39)(32 37)(41 62)(42 50)(43 64)(44 52)(45 58)(47 60)(49 54)(51 56)(53 61)(55 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), (1,13,59,28)(2,14,60,25)(3,15,57,26)(4,16,58,27)(5,36,45,22)(6,33,46,23)(7,34,47,24)(8,35,48,21)(9,55,32,44)(10,56,29,41)(11,53,30,42)(12,54,31,43)(17,63,39,52)(18,64,40,49)(19,61,37,50)(20,62,38,51), (1,61,53,48)(2,64,54,47)(3,63,55,46)(4,62,56,45)(5,58,51,41)(6,57,52,44)(7,60,49,43)(8,59,50,42)(9,33,26,17)(10,36,27,20)(11,35,28,19)(12,34,25,18)(13,37,30,21)(14,40,31,24)(15,39,32,23)(16,38,29,22), (1,48)(2,7)(3,46)(4,5)(6,57)(8,59)(9,19)(10,40)(11,17)(12,38)(13,23)(14,36)(15,21)(16,34)(18,29)(20,31)(22,25)(24,27)(26,35)(28,33)(30,39)(32,37)(41,62)(42,50)(43,64)(44,52)(45,58)(47,60)(49,54)(51,56)(53,61)(55,63)>;

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A4B4C4D4E···4P4Q···4U
order12···222244444···44···4
size11···188822224···48···8

32 irreducible representations

dim111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC4232D4C4×C4⋊C4C24.C22C24.3C22C232D4C23⋊Q8C23.10D4C23.78C23C23.Q8C2×C4.4D4C2×C422C2C42C2×C4C22C22
# reps112211312114831

Matrix representation of C4232D4 in GL8(𝔽5)

41000000
01000000
00100000
00010000
00000010
00000001
00004000
00000400
,
30000000
03000000
00400000
00040000
00000100
00001000
00000001
00000010
,
23000000
03000000
00330000
00020000
00001000
00000100
00000040
00000004
,
32000000
12000000
00330000
00420000
00001000
00000400
00000040
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;

C4232D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{32}D_4
% in TeX

G:=Group("C4^2:32D4");
// GroupNames label

G:=SmallGroup(128,1394);
// by ID

G=gap.SmallGroup(128,1394);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,185,136]);
// 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^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽