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G = C42.406D4order 128 = 27

39th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.406D4, C42.144C23, C4.27C4≀C2, C4⋊Q8.15C4, C42.85(C2×C4), C42⋊C2.3C4, (C22×C4).224D4, (C4×M4(2)).18C2, C8⋊C4.144C22, C4.23(C4.10D4), C23.57(C22⋊C4), (C2×C42).188C22, C42.C2.93C22, C42(C42.2C22), C42.2C2216C2, C23.37C23.9C2, C2.31(C2×C4≀C2), C4⋊C4.24(C2×C4), (C2×C4).1172(C2×D4), (C22×C4).210(C2×C4), (C2×C4).138(C22×C4), C2.10(C2×C4.10D4), (C2×C4).244(C22⋊C4), C22.202(C2×C22⋊C4), (C2×C4)(C42.2C22), SmallGroup(128,258)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.406D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.406D4
C1C22C2×C4 — C42.406D4
C1C2×C4C2×C42 — C42.406D4
C1C22C22C42 — C42.406D4

Generators and relations for C42.406D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 196 in 113 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×7], C22, C22 [×3], C8 [×8], 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], M4(2) [×8], C22×C4, C22×C4 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C42.2C22 [×4], C4×M4(2) [×2], C23.37C23, C42.406D4
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], C4≀C2 [×4], C2×C22⋊C4, C2×C4.10D4, C2×C4≀C2 [×2], C42.406D4

Smallest permutation representation of C42.406D4
On 64 points
Generators in S64
(1 22 60 16)(2 23 61 9)(3 24 62 10)(4 17 63 11)(5 18 64 12)(6 19 57 13)(7 20 58 14)(8 21 59 15)(25 48 52 38)(26 41 53 39)(27 42 54 40)(28 43 55 33)(29 44 56 34)(30 45 49 35)(31 46 50 36)(32 47 51 37)
(1 20 64 10)(2 17 57 15)(3 22 58 12)(4 19 59 9)(5 24 60 14)(6 21 61 11)(7 18 62 16)(8 23 63 13)(25 40 56 46)(26 37 49 43)(27 34 50 48)(28 39 51 45)(29 36 52 42)(30 33 53 47)(31 38 54 44)(32 35 55 41)
(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 38 20 54 64 44 10 31)(2 26 17 37 57 49 15 43)(3 46 22 25 58 40 12 56)(4 51 19 45 59 28 9 39)(5 34 24 50 60 48 14 27)(6 30 21 33 61 53 11 47)(7 42 18 29 62 36 16 52)(8 55 23 41 63 32 13 35)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111112224
type++++++-
imageC1C2C2C2C4C4D4D4C4≀C2C4.10D4
kernelC42.406D4C42.2C22C4×M4(2)C23.37C23C42⋊C2C4⋊Q8C42C22×C4C4C4
# reps14214422162

Matrix representation of C42.406D4 in GL4(𝔽17) generated by

4000
0400
0040
0004
,
1000
01600
00130
00013
,
0400
1000
0040
0001
,
1000
0400
00016
0040
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[0,1,0,0,4,0,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,0,4,0,0,16,0] >;

C42.406D4 in GAP, Magma, Sage, TeX

C_4^2._{406}D_4
% in TeX

G:=Group("C4^2.406D4");
// GroupNames label

G:=SmallGroup(128,258);
// by ID

G=gap.SmallGroup(128,258);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,352,1123,1018,248,1971,102]);
// 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=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations

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