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G = C42.262D4order 128 = 27

244th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.262D4, C42.727C23, C83Q88C2, C4⋊Q1621C2, C8.14(C4○D4), C8.D4.5C2, Q16⋊C418C2, C4⋊C4.124C23, (C4×C8).188C22, (C2×C8).464C23, (C2×C4).383C24, C4.SD1645C2, C23.270(C2×D4), (C22×C4).481D4, (C4×M4(2)).6C2, C4⋊Q8.298C22, (C4×Q8).99C22, C4.Q8.34C22, C4.24(C8.C22), (C2×Q16).68C22, (C2×Q8).124C23, C8⋊C4.140C22, (C2×C42).869C22, C22.643(C22×D4), C22⋊Q8.184C22, (C22×C4).1061C23, Q8⋊C4.133C22, (C2×M4(2)).291C22, C23.37C23.36C2, C2.80(C22.26C24), C4.68(C2×C4○D4), (C2×C4).1226(C2×D4), C2.48(C2×C8.C22), SmallGroup(128,1917)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.262D4
C1C2C4C2×C4C42C8⋊C4C4×M4(2) — C42.262D4
C1C2C2×C4 — C42.262D4
C1C22C2×C42 — C42.262D4
C1C2C2C2×C4 — C42.262D4

Generators and relations for C42.262D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=a2c3 >

Subgroups: 292 in 179 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×11], C22, C22 [×3], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×12], C23, C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], M4(2) [×4], Q16 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], Q8⋊C4 [×8], C4.Q8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C2×M4(2) [×2], C2×Q16 [×4], C4×M4(2), Q16⋊C4 [×4], C8.D4 [×4], C4.SD16 [×2], C4⋊Q16, C83Q8, C23.37C23 [×2], C42.262D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×4], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8.C22 [×2], C42.262D4

Smallest permutation representation of C42.262D4
On 64 points
Generators in S64
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 55 21 51)(18 52 22 56)(19 49 23 53)(20 54 24 50)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 59 45 63)(42 64 46 60)(43 61 47 57)(44 58 48 62)
(1 39 31 10)(2 36 32 15)(3 33 25 12)(4 38 26 9)(5 35 27 14)(6 40 28 11)(7 37 29 16)(8 34 30 13)(17 64 49 48)(18 61 50 45)(19 58 51 42)(20 63 52 47)(21 60 53 44)(22 57 54 41)(23 62 55 46)(24 59 56 43)
(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 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 54 13 50)(10 53 14 49)(11 52 15 56)(12 51 16 55)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(25 58 29 62)(26 57 30 61)(27 64 31 60)(28 63 32 59)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A···4J4K4L···4S8A···8H
order122224···444···48···8
size111142···248···84···4

32 irreducible representations

dim111111112224
type++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D4C8.C22
kernelC42.262D4C4×M4(2)Q16⋊C4C8.D4C4.SD16C4⋊Q16C83Q8C23.37C23C42C22×C4C8C4
# reps114421122284

Matrix representation of C42.262D4 in GL6(𝔽17)

1600000
010000
000100
0016000
00134162
0004161
,
400000
040000
001000
000100
0000160
00413016
,
1300000
040000
000010
00413115
0001600
0001604
,
040000
1300000
005500
0051200
0031407
0003120

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,13,0,0,0,1,0,4,4,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,13,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,13,16,16,0,0,1,1,0,0,0,0,0,15,0,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,5,5,3,0,0,0,5,12,14,3,0,0,0,0,0,12,0,0,0,0,7,0] >;

C42.262D4 in GAP, Magma, Sage, TeX

C_4^2._{262}D_4
% in TeX

G:=Group("C4^2.262D4");
// GroupNames label

G:=SmallGroup(128,1917);
// by ID

G=gap.SmallGroup(128,1917);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,520,521,80,4037,124]);
// Polycyclic

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

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