Copied to
clipboard

G = C42.165D4order 128 = 27

147th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.165D4, C23.436C24, C24.319C23, C22.1732- 1+4, (C2×Q8)⋊24D4, C428C441C2, C4.166(C4⋊D4), C2.36(Q85D4), C23.50(C4○D4), (C22×C4).93C23, C23.7Q864C2, (C23×C4).389C22, (C2×C42).542C22, C22.287(C22×D4), C24.C2277C2, C23.10D4.18C2, (C22×D4).527C22, (C22×Q8).431C22, C23.81C2335C2, C2.59(C22.19C24), C2.C42.179C22, C2.20(C23.38C23), C2.60(C22.46C24), (C2×C4×Q8)⋊21C2, (C2×C4×D4).58C2, C2.31(C2×C4⋊D4), (C2×C22⋊Q8)⋊19C2, (C2×C4).1194(C2×D4), (C2×C4).818(C4○D4), (C2×C4⋊C4).296C22, C22.313(C2×C4○D4), (C2×C22⋊C4).172C22, SmallGroup(128,1268)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.165D4
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C42.165D4
C1C23 — C42.165D4
C1C23 — C42.165D4
C1C23 — C42.165D4

Generators and relations for C42.165D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 548 in 302 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×16], 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 [×20], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C4×D4 [×4], C4×Q8 [×4], C22⋊Q8 [×8], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8 [×2], C428C4, C24.C22 [×4], C23.10D4 [×2], C23.81C23 [×2], C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8 [×2], C42.165D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×2], C2×C4⋊D4, C22.19C24, C23.38C23, Q85D4 [×2], C22.46C24 [×2], C42.165D4

Smallest permutation representation of C42.165D4
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 11 14 25)(2 12 15 26)(3 9 16 27)(4 10 13 28)(5 39 22 30)(6 40 23 31)(7 37 24 32)(8 38 21 29)(17 36 62 51)(18 33 63 52)(19 34 64 49)(20 35 61 50)(41 53 60 46)(42 54 57 47)(43 55 58 48)(44 56 59 45)
(1 43 21 36)(2 57 22 50)(3 41 23 34)(4 59 24 52)(5 35 15 42)(6 49 16 60)(7 33 13 44)(8 51 14 58)(9 46 31 19)(10 56 32 63)(11 48 29 17)(12 54 30 61)(18 28 45 37)(20 26 47 39)(25 55 38 62)(27 53 40 64)
(1 57 14 42)(2 43 15 58)(3 59 16 44)(4 41 13 60)(5 51 22 36)(6 33 23 52)(7 49 24 34)(8 35 21 50)(9 56 27 45)(10 46 28 53)(11 54 25 47)(12 48 26 55)(17 39 62 30)(18 31 63 40)(19 37 64 32)(20 29 61 38)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111122224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42- 1+4
kernelC42.165D4C23.7Q8C428C4C24.C22C23.10D4C23.81C23C2×C4×D4C2×C4×Q8C2×C22⋊Q8C42C2×Q8C2×C4C23C22
# reps12142211244482

Matrix representation of C42.165D4 in GL6(𝔽5)

130000
140000
000400
001000
000040
000041
,
100000
010000
001000
000100
000020
000023
,
200000
230000
000100
001000
000042
000041
,
340000
320000
001000
000100
000013
000014

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

C42.165D4 in GAP, Magma, Sage, TeX

C_4^2._{165}D_4
% in TeX

G:=Group("C4^2.165D4");
// GroupNames label

G:=SmallGroup(128,1268);
// by ID

G=gap.SmallGroup(128,1268);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,675,80]);
// 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*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

׿
×
𝔽