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G = C42.268D4order 128 = 27

250th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.268D4, C42.729C23, C4.1022+ 1+4, C85D423C2, C4⋊Q169C2, D4.7D43C2, C4.29(C4○D8), C4.4D812C2, Q8.D41C2, C4⋊C4.149C23, C4⋊C8.312C22, (C2×C8).326C23, (C4×C8).265C22, (C2×C4).408C24, C4.SD1626C2, C23.285(C2×D4), (C22×C4).498D4, C4⋊Q8.302C22, D4⋊C4.1C22, (C2×D4).157C23, C4.27(C8.C22), (C2×Q8).145C23, (C4×Q8).101C22, (C2×Q16).26C22, C42.12C434C2, C41D4.163C22, C22⋊C8.193C22, (C2×C42).875C22, (C2×SD16).85C22, C22.668(C22×D4), C22⋊Q8.193C22, (C22×C4).1079C23, Q8⋊C4.100C22, C4.4D4.150C22, C23.37C2317C2, C2.79(C22.29C24), C22.26C24.41C2, C2.42(C2×C4○D8), (C2×C4).538(C2×D4), C2.55(C2×C8.C22), (C2×C4○D4).172C22, SmallGroup(128,1942)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.268D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.268D4
C1C2C2×C4 — C42.268D4
C1C22C2×C42 — C42.268D4
C1C2C2C2×C4 — C42.268D4

Generators and relations for C42.268D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c3 >

Subgroups: 396 in 197 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C4.4D4 [×2], C42.C2, C41D4, C4⋊Q8 [×3], C2×SD16 [×4], C2×Q16 [×4], C2×C4○D4 [×2], C42.12C4, D4.7D4 [×4], Q8.D4 [×4], C4.4D8, C4.SD16, C85D4, C4⋊Q16, C22.26C24, C23.37C23, C42.268D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C4○D8, C2×C8.C22, C42.268D4

Smallest permutation representation of C42.268D4
On 64 points
Generators in S64
(1 46 51 59)(2 47 52 60)(3 48 53 61)(4 41 54 62)(5 42 55 63)(6 43 56 64)(7 44 49 57)(8 45 50 58)(9 24 29 37)(10 17 30 38)(11 18 31 39)(12 19 32 40)(13 20 25 33)(14 21 26 34)(15 22 27 35)(16 23 28 36)
(1 14 5 10)(2 27 6 31)(3 16 7 12)(4 29 8 25)(9 50 13 54)(11 52 15 56)(17 46 21 42)(18 60 22 64)(19 48 23 44)(20 62 24 58)(26 55 30 51)(28 49 32 53)(33 41 37 45)(34 63 38 59)(35 43 39 47)(36 57 40 61)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 33 21 37)(18 36 22 40)(19 39 23 35)(20 34 24 38)(25 26 29 30)(27 32 31 28)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4J4K4L···4Q8A···8H
order12222224···444···48···8
size11114882···248···84···4

32 irreducible representations

dim111111111122244
type++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D8C8.C222+ 1+4
kernelC42.268D4C42.12C4D4.7D4Q8.D4C4.4D8C4.SD16C85D4C4⋊Q16C22.26C24C23.37C23C42C22×C4C4C4C4
# reps114411111122822

Matrix representation of C42.268D4 in GL6(𝔽17)

100000
010000
000100
0016000
000001
0000160
,
400000
040000
000010
000001
0016000
0001600
,
1250000
12120000
0000125
00001212
0012500
00121200
,
1250000
550000
0000125
000055
0012500
005500

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,12,12,0,0,0,0,5,12,0,0],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,5,0,0,0,0,5,5,0,0] >;

C42.268D4 in GAP, Magma, Sage, TeX

C_4^2._{268}D_4
% in TeX

G:=Group("C4^2.268D4");
// GroupNames label

G:=SmallGroup(128,1942);
// by ID

G=gap.SmallGroup(128,1942);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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