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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 [×3], C2 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×13], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C4×C8, C8⋊C4 [×4], C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C2×M4(2), C42.2C22 [×4], C4×M4(2), C42.6C4, C2×C42.C2, C42.408D4
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 [×2], 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 58 20 12)(2 13 21 59)(3 60 22 14)(4 15 23 61)(5 62 24 16)(6 9 17 63)(7 64 18 10)(8 11 19 57)(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 10 24 60)(2 15 17 57)(3 12 18 62)(4 9 19 59)(5 14 20 64)(6 11 21 61)(7 16 22 58)(8 13 23 63)(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 10 54 24 44 60 31)(2 26 15 37 17 49 57 43)(3 46 12 25 18 40 62 56)(4 51 9 45 19 28 59 39)(5 34 14 50 20 48 64 27)(6 30 11 33 21 53 61 47)(7 42 16 29 22 36 58 52)(8 55 13 41 23 32 63 35)

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

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

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

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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