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G = C42.69D4order 128 = 27

51st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.69D4, C42.150C23, (C4×D4).4C4, (C4×Q8).4C4, C22.5C4≀C2, C4⋊D4.11C4, C42.91(C2×C4), C22⋊Q8.11C4, (C22×C4).666D4, C8⋊C4.86C22, C42.6C436C2, C42.2C229C2, (C2×C42).194C22, C42.C2.97C22, C23.106(C22⋊C4), C42.C2210C2, C4.4D4.116C22, C2.32(C42⋊C22), C23.36C23.11C2, C2.12(M4(2).8C22), C2.37(C2×C4≀C2), C4⋊C4.28(C2×C4), (C2×C8⋊C4)⋊16C2, (C2×D4).23(C2×C4), (C2×Q8).23(C2×C4), (C2×C4).1178(C2×D4), (C2×C4).144(C22×C4), (C22×C4).216(C2×C4), (C2×C4).322(C22⋊C4), C22.208(C2×C22⋊C4), SmallGroup(128,264)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.69D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.69D4
C1C22C2×C4 — C42.69D4
C1C22C2×C42 — C42.69D4
C1C22C22C42 — C42.69D4

Generators and relations for C42.69D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=a2bc3 >

Subgroups: 212 in 109 conjugacy classes, 44 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×9], C22, C22 [×2], C22 [×5], C8 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C8⋊C4 [×4], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22×C8, C42.C22 [×2], C42.2C22 [×2], C2×C8⋊C4, C42.6C4, C23.36C23, C42.69D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, M4(2).8C22, C2×C4≀C2, C42⋊C22, C42.69D4

Smallest permutation representation of C42.69D4
On 64 points
Generators in S64
(1 44 5 48)(2 29 6 25)(3 46 7 42)(4 31 8 27)(9 49 13 53)(10 59 14 63)(11 51 15 55)(12 61 16 57)(17 26 21 30)(18 43 22 47)(19 28 23 32)(20 45 24 41)(33 54 37 50)(34 64 38 60)(35 56 39 52)(36 58 40 62)
(1 56 19 57)(2 53 20 62)(3 50 21 59)(4 55 22 64)(5 52 23 61)(6 49 24 58)(7 54 17 63)(8 51 18 60)(9 45 36 29)(10 42 37 26)(11 47 38 31)(12 44 39 28)(13 41 40 25)(14 46 33 30)(15 43 34 27)(16 48 35 32)
(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 51 56 18 19 60 57 8)(2 21 53 59 20 3 62 50)(4 5 55 52 22 23 64 61)(6 17 49 63 24 7 58 54)(9 33 45 30 36 14 29 46)(10 25 42 13 37 41 26 40)(11 16 47 48 38 35 31 32)(12 43 44 34 39 27 28 15)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K4L4M8A···8H8I8J8K8L
order12222224···4444448···88888
size11112282···2448884···48888

32 irreducible representations

dim111111111122244
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2M4(2).8C22C42⋊C22
kernelC42.69D4C42.C22C42.2C22C2×C8⋊C4C42.6C4C23.36C23C4×D4C4×Q8C4⋊D4C22⋊Q8C42C22×C4C22C2C2
# reps122111222222822

Matrix representation of C42.69D4 in GL6(𝔽17)

1300000
0130000
0040150
0000161
00160130
00161130
,
420000
1130000
004000
000400
000040
000004
,
1230000
1050000
000298
000208
000800
0080015
,
1230000
1000000
000298
000090
000800
009820

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

C42.69D4 in GAP, Magma, Sage, TeX

C_4^2._{69}D_4
% in TeX

G:=Group("C4^2.69D4");
// GroupNames label

G:=SmallGroup(128,264);
// by ID

G=gap.SmallGroup(128,264);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,520,1123,1018,248,1971,102]);
// Polycyclic

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

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