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G = C42.416D4order 128 = 27

49th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.416D4, C42.167C23, C4⋊Q8.24C4, (C2×C4).13Q16, C4.37(C2×Q16), C4.55(C2×SD16), (C2×C4).30SD16, C4.10D834C2, C4⋊C8.205C22, C42.108(C2×C4), (C22×C4).239D4, C4⋊Q8.240C22, C4.109(C8⋊C22), C4.18(Q8⋊C4), C4.7(C4.10D4), C4⋊M4(2).16C2, (C2×C42).211C22, C23.184(C22⋊C4), C42.12C4.26C2, C22.23(Q8⋊C4), C2.12(C23.37D4), (C2×C4⋊Q8).5C2, (C2×C4⋊C4).21C4, C4⋊C4.38(C2×C4), (C2×C4).1238(C2×D4), C2.15(C2×Q8⋊C4), (C22×C4).233(C2×C4), (C2×C4).161(C22×C4), C2.18(C2×C4.10D4), (C2×C4).248(C22⋊C4), C22.225(C2×C22⋊C4), SmallGroup(128,281)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.416D4
C1C2C22C2×C4C42C2×C42C2×C4⋊Q8 — C42.416D4
C1C22C2×C4 — C42.416D4
C1C22C2×C42 — C42.416D4
C1C22C22C42 — C42.416D4

Generators and relations for C42.416D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 236 in 120 conjugacy classes, 56 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×12], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×8], C4×C8, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2), C22×Q8, C4.10D8 [×4], C4⋊M4(2), C42.12C4, C2×C4⋊Q8, C42.416D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C4.10D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C8⋊C22 [×2], C2×C4.10D4, C2×Q8⋊C4, C23.37D4, C42.416D4

Smallest permutation representation of C42.416D4
On 64 points
Generators in S64
(1 20 64 16)(2 9 57 21)(3 22 58 10)(4 11 59 23)(5 24 60 12)(6 13 61 17)(7 18 62 14)(8 15 63 19)(25 41 52 40)(26 33 53 42)(27 43 54 34)(28 35 55 44)(29 45 56 36)(30 37 49 46)(31 47 50 38)(32 39 51 48)
(1 14 60 22)(2 23 61 15)(3 16 62 24)(4 17 63 9)(5 10 64 18)(6 19 57 11)(7 12 58 20)(8 21 59 13)(25 38 56 43)(26 44 49 39)(27 40 50 45)(28 46 51 33)(29 34 52 47)(30 48 53 35)(31 36 54 41)(32 42 55 37)
(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 48 14 53 60 35 22 30)(2 52 23 47 61 29 15 34)(3 46 16 51 62 33 24 28)(4 50 17 45 63 27 9 40)(5 44 10 49 64 39 18 26)(6 56 19 43 57 25 11 38)(7 42 12 55 58 37 20 32)(8 54 21 41 59 31 13 36)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222244
type+++++++--+
imageC1C2C2C2C2C4C4D4D4SD16Q16C4.10D4C8⋊C22
kernelC42.416D4C4.10D8C4⋊M4(2)C42.12C4C2×C4⋊Q8C2×C4⋊C4C4⋊Q8C42C22×C4C2×C4C2×C4C4C4
# reps1411144224422

Matrix representation of C42.416D4 in GL6(𝔽17)

0160000
100000
0016000
0001600
0000160
0000016
,
0160000
100000
00161600
002100
0051601
0076160
,
010000
100000
0011209
000288
0001234
0013111211
,
550000
1250000
0045150
000922
0001615
0011273

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

C42.416D4 in GAP, Magma, Sage, TeX

C_4^2._{416}D_4
% in TeX

G:=Group("C4^2.416D4");
// GroupNames label

G:=SmallGroup(128,281);
// by ID

G=gap.SmallGroup(128,281);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,680,758,1123,1018,248,1971,242]);
// 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=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations

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