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G = C42.68D4order 128 = 27

50th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.68D4, C42.149C23, C4.16C4≀C2, C4⋊Q8.17C4, C42.90(C2×C4), C42⋊C2.4C4, (C22×C4).739D4, C8⋊C4.147C22, C42.6C4.20C2, C42.2C228C2, (C2×C42).193C22, C42.C2.96C22, C23.105(C22⋊C4), C22.2(C4.10D4), C2.31(C42⋊C22), C23.37C23.10C2, C2.36(C2×C4≀C2), C4⋊C4.27(C2×C4), (C2×C8⋊C4).20C2, (C2×C4).1177(C2×D4), (C22×C4).215(C2×C4), (C2×C4).143(C22×C4), C2.12(C2×C4.10D4), (C2×C4).321(C22⋊C4), C22.207(C2×C22⋊C4), SmallGroup(128,263)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.68D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.68D4
C1C22C2×C4 — C42.68D4
C1C22C2×C42 — C42.68D4
C1C22C22C42 — C42.68D4

Generators and relations for C42.68D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, 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 [×9], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C22×C4 [×3], C2×Q8 [×2], C8⋊C4 [×4], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C22×C8, C42.2C22 [×4], C2×C8⋊C4, C42.6C4, C23.37C23, C42.68D4
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, C42.68D4

Smallest permutation representation of C42.68D4
On 64 points
Generators in S64
(1 62 19 11)(2 16 20 59)(3 64 21 13)(4 10 22 61)(5 58 23 15)(6 12 24 63)(7 60 17 9)(8 14 18 57)(25 33 49 44)(26 41 50 38)(27 35 51 46)(28 43 52 40)(29 37 53 48)(30 45 54 34)(31 39 55 42)(32 47 56 36)
(1 60 23 13)(2 57 24 10)(3 62 17 15)(4 59 18 12)(5 64 19 9)(6 61 20 14)(7 58 21 11)(8 63 22 16)(25 39 53 46)(26 36 54 43)(27 33 55 48)(28 38 56 45)(29 35 49 42)(30 40 50 47)(31 37 51 44)(32 34 52 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 37 60 51 23 44 13 31)(2 26 57 36 24 54 10 43)(3 46 62 25 17 39 15 53)(4 56 59 45 18 28 12 38)(5 33 64 55 19 48 9 27)(6 30 61 40 20 50 14 47)(7 42 58 29 21 35 11 49)(8 52 63 41 22 32 16 34)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111122244
type+++++++-
imageC1C2C2C2C2C4C4D4D4C4≀C2C4.10D4C42⋊C22
kernelC42.68D4C42.2C22C2×C8⋊C4C42.6C4C23.37C23C42⋊C2C4⋊Q8C42C22×C4C4C22C2
# reps141114422822

Matrix representation of C42.68D4 in GL6(𝔽17)

1300000
0130000
004000
00151300
000040
00001513
,
100000
0160000
0013000
0001300
0000130
0000013
,
080000
800000
000053
00001312
0031200
0061400
,
100000
0130000
0014500
0011300
000053
00001312

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

C42.68D4 in GAP, Magma, Sage, TeX

C_4^2._{68}D_4
% in TeX

G:=Group("C4^2.68D4");
// GroupNames label

G:=SmallGroup(128,263);
// by ID

G=gap.SmallGroup(128,263);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,352,1123,1018,248,1971,102]);
// 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,c*a*c^-1=d*a*d^-1=a*b^2,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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