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G = C42.408D4order 128 = 27

41st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.408D4, C42.146C23, C22.26C4≀C2, C42.87(C2×C4), (C22×C4).226D4, C42.C2.10C4, C4.5(C4.10D4), (C4×M4(2)).19C2, C8⋊C4.145C22, C42.6C4.19C2, (C2×C42).190C22, C42.2C226C2, C42.C2.94C22, C23.178(C22⋊C4), C2.28(C42⋊C22), C2.33(C2×C4≀C2), (C2×C4⋊C4).17C4, C4⋊C4.25(C2×C4), (C2×C4).1174(C2×D4), (C2×C42.C2).3C2, (C22×C4).212(C2×C4), (C2×C4).140(C22×C4), C2.11(C2×C4.10D4), (C2×C4).178(C22⋊C4), C22.204(C2×C22⋊C4), SmallGroup(128,260)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.408D4
C1C2C22C2×C4C42C2×C42C2×C42.C2 — C42.408D4
C1C22C2×C4 — C42.408D4
C1C22C2×C42 — C42.408D4
C1C22C22C42 — C42.408D4

Generators and relations for C42.408D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 188 in 106 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C2×M4(2), C42.2C22, C4×M4(2), C42.6C4, C2×C42.C2, C42.408D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C4≀C2, C2×C22⋊C4, C2×C4.10D4, C2×C4≀C2, C42⋊C22, C42.408D4

Smallest permutation representation of C42.408D4
On 64 points
Generators in S64
(1 16 60 22)(2 23 61 9)(3 10 62 24)(4 17 63 11)(5 12 64 18)(6 19 57 13)(7 14 58 20)(8 21 59 15)(25 38 52 48)(26 41 53 39)(27 40 54 42)(28 43 55 33)(29 34 56 44)(30 45 49 35)(31 36 50 46)(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)| (1,16,60,22)(2,23,61,9)(3,10,62,24)(4,17,63,11)(5,12,64,18)(6,19,57,13)(7,14,58,20)(8,21,59,15)(25,38,52,48)(26,41,53,39)(27,40,54,42)(28,43,55,33)(29,34,56,44)(30,45,49,35)(31,36,50,46)(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( (1,16,60,22)(2,23,61,9)(3,10,62,24)(4,17,63,11)(5,12,64,18)(6,19,57,13)(7,14,58,20)(8,21,59,15)(25,38,52,48)(26,41,53,39)(27,40,54,42)(28,43,55,33)(29,34,56,44)(30,45,49,35)(31,36,50,46)(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([[(1,16,60,22),(2,23,61,9),(3,10,62,24),(4,17,63,11),(5,12,64,18),(6,19,57,13),(7,14,58,20),(8,21,59,15),(25,38,52,48),(26,41,53,39),(27,40,54,42),(28,43,55,33),(29,34,56,44),(30,45,49,35),(31,36,50,46),(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)]])

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim111111122244
type+++++++-
imageC1C2C2C2C2C4C4D4D4C4≀C2C4.10D4C42⋊C22
kernelC42.408D4C42.2C22C4×M4(2)C42.6C4C2×C42.C2C2×C4⋊C4C42.C2C42C22×C4C22C4C2
# reps141114422822

Matrix representation of C42.408D4 in GL6(𝔽17)

190000
13160000
0016000
0001600
0000160
0000016
,
420000
1130000
000100
0016000
0021142
001011013
,
5100000
9120000
0014327
0014697
0067114
0089413
,
0120000
630000
0002160
001361315
00911913
0013962

G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,9,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,0,16,2,10,0,0,1,0,11,11,0,0,0,0,4,0,0,0,0,0,2,13],[5,9,0,0,0,0,10,12,0,0,0,0,0,0,14,14,6,8,0,0,3,6,7,9,0,0,2,9,1,4,0,0,7,7,14,13],[0,6,0,0,0,0,12,3,0,0,0,0,0,0,0,13,9,13,0,0,2,6,11,9,0,0,16,13,9,6,0,0,0,15,13,2] >;

C42.408D4 in GAP, Magma, Sage, TeX

C_4^2._{408}D_4
% in TeX

G:=Group("C4^2.408D4");
// GroupNames label

G:=SmallGroup(128,260);
// by ID

G=gap.SmallGroup(128,260);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,1123,1018,248,1971,102]);
// 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,c*a*c^-1=a^-1,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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