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G = C42.412D4order 128 = 27

45th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.412D4, C42.162C23, C4.98(C4○D8), C43(C4.10D8), C4.10D845C2, C4⋊C8.258C22, C42⋊C2.6C4, C42.103(C2×C4), (C22×C4).235D4, C4⋊Q8.236C22, C42.C2.12C4, C4.24(C4.10D4), C23.63(C22⋊C4), (C2×C42).206C22, C42.12C4.23C2, C2.15(C23.24D4), C23.37C23.13C2, C4⋊C4.35(C2×C4), (C2×C4)(C4.10D8), (C2×C4).1233(C2×D4), (C2×C4).156(C22×C4), (C22×C4).228(C2×C4), C2.16(C2×C4.10D4), (C2×C4).181(C22⋊C4), C22.220(C2×C22⋊C4), SmallGroup(128,276)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.412D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.412D4
C1C22C2×C4 — C42.412D4
C1C2×C4C2×C42 — C42.412D4
C1C22C22C42 — C42.412D4

Generators and relations for C42.412D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 180 in 101 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×7], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×2], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4.10D8 [×4], C42.12C4 [×2], C23.37C23, C42.412D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C4○D8 [×4], C2×C4.10D4, C23.24D4 [×2], C42.412D4

Smallest permutation representation of C42.412D4
On 64 points
Generators in S64
(1 35 54 63)(2 36 55 64)(3 37 56 57)(4 38 49 58)(5 39 50 59)(6 40 51 60)(7 33 52 61)(8 34 53 62)(9 23 27 43)(10 24 28 44)(11 17 29 45)(12 18 30 46)(13 19 31 47)(14 20 32 48)(15 21 25 41)(16 22 26 42)
(1 61 50 37)(2 38 51 62)(3 63 52 39)(4 40 53 64)(5 57 54 33)(6 34 55 58)(7 59 56 35)(8 36 49 60)(9 21 31 45)(10 46 32 22)(11 23 25 47)(12 48 26 24)(13 17 27 41)(14 42 28 18)(15 19 29 43)(16 44 30 20)
(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 61 26 50 24 37 12)(2 15 38 19 51 29 62 43)(3 46 63 32 52 22 39 10)(4 13 40 17 53 27 64 41)(5 44 57 30 54 20 33 16)(6 11 34 23 55 25 58 47)(7 42 59 28 56 18 35 14)(8 9 36 21 49 31 60 45)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4L4M4N4O4P4Q8A···8P
order1222244444···4444448···8
size1111411112···2488884···4

38 irreducible representations

dim1111112224
type++++++-
imageC1C2C2C2C4C4D4D4C4○D8C4.10D4
kernelC42.412D4C4.10D8C42.12C4C23.37C23C42⋊C2C42.C2C42C22×C4C4C4
# reps14214422162

Matrix representation of C42.412D4 in GL4(𝔽17) generated by

4000
0400
00130
00013
,
1000
0100
00130
0004
,
8000
0200
0002
0080
,
01500
8000
0080
0002
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[8,0,0,0,0,2,0,0,0,0,0,8,0,0,2,0],[0,8,0,0,15,0,0,0,0,0,8,0,0,0,0,2] >;

C42.412D4 in GAP, Magma, Sage, TeX

C_4^2._{412}D_4
% in TeX

G:=Group("C4^2.412D4");
// GroupNames label

G:=SmallGroup(128,276);
// by ID

G=gap.SmallGroup(128,276);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,352,1123,1018,248,1971,242]);
// Polycyclic

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

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