Copied to
clipboard

G = C4223D4order 128 = 27

17th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4223D4, C24.352C23, C23.501C24, C22.2062- 1+4, C22.2822+ 1+4, C23.Q834C2, C23.8Q878C2, C23.161(C4○D4), (C23×C4).412C22, (C2×C42).588C22, (C22×C4).552C23, C22.331(C22×D4), C24.C2299C2, C23.10D4.29C2, C23.23D4.43C2, (C22×D4).184C22, C23.81C2352C2, C23.65C2397C2, C2.74(C22.19C24), C24.3C22.55C2, C23.63C23105C2, C2.67(C22.45C24), C2.C42.231C22, C2.44(C22.26C24), C2.51(C22.50C24), C2.74(C22.46C24), C2.77(C22.47C24), C2.19(C22.31C24), (C4×C4⋊C4)⋊111C2, (C2×C4).369(C2×D4), (C2×C422C2)⋊13C2, (C2×C42⋊C2)⋊33C2, (C2×C4).409(C4○D4), (C2×C4⋊C4).341C22, C22.377(C2×C4○D4), (C2×C22⋊C4).515C22, SmallGroup(128,1333)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4223D4
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C4223D4
C1C23 — C4223D4
C1C23 — C4223D4
C1C23 — C4223D4

Generators and relations for C4223D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 484 in 259 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×41], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×19], C22×C4 [×13], C22×C4 [×6], C2×D4 [×5], C24 [×2], C2.C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×10], C42⋊C2 [×4], C422C2 [×4], C23×C4, C22×D4, C4×C4⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22 [×3], C23.65C23, C24.3C22, C23.10D4, C23.Q8 [×2], C23.81C23, C2×C42⋊C2, C2×C422C2, C4223D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C22.19C24, C22.26C24, C22.31C24, C22.45C24, C22.46C24, C22.47C24, C22.50C24, C4223D4

Smallest permutation representation of C4223D4
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 16 59 27)(2 13 60 28)(3 14 57 25)(4 15 58 26)(5 33 45 21)(6 34 46 22)(7 35 47 23)(8 36 48 24)(9 56 32 41)(10 53 29 42)(11 54 30 43)(12 55 31 44)(17 64 37 49)(18 61 38 50)(19 62 39 51)(20 63 40 52)
(1 48 53 61)(2 47 54 64)(3 46 55 63)(4 45 56 62)(5 41 51 58)(6 44 52 57)(7 43 49 60)(8 42 50 59)(9 17 26 35)(10 20 27 34)(11 19 28 33)(12 18 25 36)(13 21 30 39)(14 24 31 38)(15 23 32 37)(16 22 29 40)
(2 58)(4 60)(5 64)(6 52)(7 62)(8 50)(9 32)(10 12)(11 30)(13 28)(14 16)(15 26)(17 23)(18 34)(19 21)(20 36)(22 38)(24 40)(25 27)(29 31)(33 39)(35 37)(41 54)(43 56)(45 49)(46 63)(47 51)(48 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC4223D4C4×C4⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.81C23C2×C42⋊C2C2×C422C2C42C2×C4C23C22C22
# reps1111131112111412411

Matrix representation of C4223D4 in GL6(𝔽5)

320000
120000
001000
000100
000003
000030
,
300000
030000
001000
000100
000004
000040
,
100000
010000
004200
004100
000001
000040
,
100000
240000
001000
001400
000010
000004

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

C4223D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{23}D_4
% in TeX

G:=Group("C4^2:23D4");
// GroupNames label

G:=SmallGroup(128,1333);
// by ID

G=gap.SmallGroup(128,1333);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675,248]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽