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G = C42.410D4order 128 = 27

43rd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.410D4, C42.160C23, (C4×Q8).6C4, (C2×C4).58Q16, C4.35(C2×Q16), C4.96(C4○D8), C4(C4.6Q16), C4.53(C2×SD16), C22⋊Q8.13C4, C42(C4.10D8), C4.10D844C2, C4⋊C8.256C22, C42.101(C2×C4), C4.6Q1627C2, (C2×C4).102SD16, (C22×C4).233D4, C4⋊Q8.234C22, C4.34(Q8⋊C4), (C2×C42).204C22, C22.7(Q8⋊C4), C23.108(C22⋊C4), C42.12C4.22C2, C2.13(C23.24D4), C23.37C23.12C2, C2.15(M4(2).8C22), (C2×C4⋊C8).13C2, C4⋊C4.34(C2×C4), (C2×Q8).28(C2×C4), (C2×C4).1231(C2×D4), C2.13(C2×Q8⋊C4), (C22×C4).226(C2×C4), (C2×C4).154(C22×C4), (C2×C4).105(C22⋊C4), C22.218(C2×C22⋊C4), SmallGroup(128,274)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.410D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.410D4
C1C22C2×C4 — C42.410D4
C1C2×C4C2×C42 — C42.410D4
C1C22C22C42 — C42.410D4

Generators and relations for C42.410D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 106 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×7], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×6], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C22×C8, C4.10D8 [×2], C4.6Q16 [×2], C2×C4⋊C8, C42.12C4, C23.37C23, C42.410D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C4○D8 [×2], M4(2).8C22, C2×Q8⋊C4, C23.24D4, C42.410D4

Smallest permutation representation of C42.410D4
On 64 points
Generators in S64
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)
(1 16 58 22)(2 23 59 9)(3 10 60 24)(4 17 61 11)(5 12 62 18)(6 19 63 13)(7 14 64 20)(8 21 57 15)(25 38 50 41)(26 42 51 39)(27 40 52 43)(28 44 53 33)(29 34 54 45)(30 46 55 35)(31 36 56 47)(32 48 49 37)
(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 48 16 49 58 37 22 32)(2 52 23 43 59 27 9 40)(3 46 10 55 60 35 24 30)(4 50 17 41 61 25 11 38)(5 44 12 53 62 33 18 28)(6 56 19 47 63 31 13 36)(7 42 14 51 64 39 20 26)(8 54 21 45 57 29 15 34)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K4L4M4N4O4P8A···8P
order12222244444···44444448···8
size11112211112···24488884···4

38 irreducible representations

dim11111111222224
type++++++++-
imageC1C2C2C2C2C2C4C4D4D4SD16Q16C4○D8M4(2).8C22
kernelC42.410D4C4.10D8C4.6Q16C2×C4⋊C8C42.12C4C23.37C23C4×Q8C22⋊Q8C42C22×C4C2×C4C2×C4C4C2
# reps12211144224482

Matrix representation of C42.410D4 in GL4(𝔽17) generated by

1000
0100
0040
0004
,
4000
01300
00160
00016
,
0900
2000
0090
0002
,
15000
0900
00015
0090
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,2,0,0,9,0,0,0,0,0,9,0,0,0,0,2],[15,0,0,0,0,9,0,0,0,0,0,9,0,0,15,0] >;

C42.410D4 in GAP, Magma, Sage, TeX

C_4^2._{410}D_4
% in TeX

G:=Group("C4^2.410D4");
// GroupNames label

G:=SmallGroup(128,274);
// by ID

G=gap.SmallGroup(128,274);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,184,1123,1018,248,1971,242]);
// Polycyclic

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

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