Copied to
clipboard

G = C42.170D4order 128 = 27

152nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.170D4, C24.325C23, C23.447C24, C22.1802- 1+4, (C2×D4).213D4, C43(C4.4D4), C23.51(C2×D4), C4.61(C4⋊D4), C2.51(D46D4), (C22×C4).96C23, C23.7Q867C2, C23.11D442C2, (C23×C4).396C22, (C2×C42).552C22, C22.298(C22×D4), (C22×D4).530C22, (C22×Q8).131C22, C23.67C2360C2, C2.C42.185C22, C2.38(C22.50C24), C2.24(C23.38C23), (C4×C4⋊C4)⋊90C2, (C2×C4⋊Q8)⋊14C2, (C2×C4×D4).61C2, (C2×C4).355(C2×D4), C2.39(C2×C4⋊D4), C2.21(C2×C4.4D4), (C2×C4).822(C4○D4), (C2×C4⋊C4).302C22, (C2×C4.4D4).25C2, C22.324(C2×C4○D4), (C2×C22⋊C4).180C22, SmallGroup(128,1279)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.170D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.170D4
C1C23 — C42.170D4
C1C23 — C42.170D4
C1C23 — C42.170D4

Generators and relations for C42.170D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 564 in 302 conjugacy classes, 116 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×14], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×12], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×6], C4×D4 [×4], C4.4D4 [×8], C4⋊Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8 [×2], C4×C4⋊C4, C23.7Q8 [×4], C23.67C23 [×2], C23.11D4 [×4], C2×C4×D4, C2×C4.4D4 [×2], C2×C4⋊Q8, C42.170D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C4.4D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×2], C2×C4⋊D4, C2×C4.4D4, C23.38C23, D46D4 [×2], C22.50C24 [×2], C42.170D4

Smallest permutation representation of C42.170D4
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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 17 39 28)(6 18 40 25)(7 19 37 26)(8 20 38 27)(9 29 47 58)(10 30 48 59)(11 31 45 60)(12 32 46 57)(21 35 50 64)(22 36 51 61)(23 33 52 62)(24 34 49 63)
(1 17 9 34)(2 27 10 62)(3 19 11 36)(4 25 12 64)(5 58 24 16)(6 32 21 44)(7 60 22 14)(8 30 23 42)(13 38 59 52)(15 40 57 50)(18 46 35 54)(20 48 33 56)(26 45 61 53)(28 47 63 55)(29 49 41 39)(31 51 43 37)
(1 5 55 39)(2 8 56 38)(3 7 53 37)(4 6 54 40)(9 24 47 49)(10 23 48 52)(11 22 45 51)(12 21 46 50)(13 20 42 27)(14 19 43 26)(15 18 44 25)(16 17 41 28)(29 63 58 34)(30 62 59 33)(31 61 60 36)(32 64 57 35)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111112224
type++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D42- 1+4
kernelC42.170D4C4×C4⋊C4C23.7Q8C23.67C23C23.11D4C2×C4×D4C2×C4.4D4C2×C4⋊Q8C42C2×D4C2×C4C22
# reps1142412144122

Matrix representation of C42.170D4 in GL6(𝔽5)

400000
040000
002000
000300
000013
000014
,
100000
010000
001000
000100
000013
000014
,
440000
210000
000100
004000
000021
000023
,
110000
040000
000400
004000
000020
000023

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

C42.170D4 in GAP, Magma, Sage, TeX

C_4^2._{170}D_4
% in TeX

G:=Group("C4^2.170D4");
// GroupNames label

G:=SmallGroup(128,1279);
// by ID

G=gap.SmallGroup(128,1279);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,268,675,80]);
// Polycyclic

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

׿
×
𝔽