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G = C42.53D4order 128 = 27

35th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.53D4, C42.617C23, D4⋊C833C2, C4⋊Q8.12C4, C4.5(C8○D4), C4⋊D4.8C4, C41D4.9C4, (C4×C8).9C22, C42.67(C2×C4), (C4×D4).8C22, C4⋊C8.201C22, (C22×C4).207D4, C4.136(C8⋊C22), C42.6C428C2, C42.12C414C2, C23.49(C22⋊C4), (C2×C42).170C22, C2.6(C23.37D4), C22.26C24.4C2, C2.13(C42⋊C22), C4⋊C4.57(C2×C4), (C2×D4).55(C2×C4), (C2×C4).1453(C2×D4), (C2×C4).83(C22⋊C4), (C2×C4).322(C22×C4), (C22×C4).192(C2×C4), C22.172(C2×C22⋊C4), C2.22((C22×C8)⋊C2), SmallGroup(128,228)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.53D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.53D4
C1C2C2×C4 — C42.53D4
C1C2×C4C2×C42 — C42.53D4
C1C22C22C42 — C42.53D4

Generators and relations for C42.53D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 276 in 123 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×6], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×12], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, D4⋊C8 [×4], C42.12C4, C42.6C4, C22.26C24, C42.53D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], C8⋊C22 [×2], (C22×C8)⋊C2, C23.37D4, C42⋊C22, C42.53D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A···8H8I8J8K8L
order122222244444444444448···88888
size111148811112222444884···48888

32 irreducible representations

dim1111111122244
type++++++++
imageC1C2C2C2C2C4C4C4D4D4C8○D4C8⋊C22C42⋊C22
kernelC42.53D4D4⋊C8C42.12C4C42.6C4C22.26C24C4⋊D4C41D4C4⋊Q8C42C22×C4C4C4C2
# reps1411142222822

Matrix representation of C42.53D4 in GL6(𝔽17)

10150000
770000
0001160
00101615
00201615
0016101
,
400000
040000
001000
000100
000010
000001
,
1210000
550000
006744
0061004
000303
0067111
,
1210000
1350000
006744
00117013
00014014
00610114

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

C42.53D4 in GAP, Magma, Sage, TeX

C_4^2._{53}D_4
% in TeX

G:=Group("C4^2.53D4");
// GroupNames label

G:=SmallGroup(128,228);
// by ID

G=gap.SmallGroup(128,228);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1059,184,1123,570,136,172]);
// 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,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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