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G = C42.414D4order 128 = 27

47th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.414D4, C42.164C23, (C2×C4).18D8, C4.56(C2×D8), C4⋊Q8.22C4, (C2×C4).27SD16, C4.76(C2×SD16), C4.10D832C2, C4⋊C8.203C22, C42.105(C2×C4), (C22×C4).237D4, C4⋊Q8.237C22, C4.21(D4⋊C4), C4.6(C4.10D4), C4.104(C8.C22), C4⋊M4(2).14C2, (C2×C42).208C22, C22.27(D4⋊C4), C23.182(C22⋊C4), C42.12C4.24C2, C2.11(C23.38D4), (C2×C4⋊Q8).3C2, (C2×C4⋊C4).20C4, C4⋊C4.36(C2×C4), (C2×C4).1235(C2×D4), C2.15(C2×D4⋊C4), (C2×C4).158(C22×C4), (C22×C4).230(C2×C4), C2.17(C2×C4.10D4), (C2×C4).246(C22⋊C4), C22.222(C2×C22⋊C4), SmallGroup(128,278)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.414D4
C1C2C22C2×C4C42C2×C42C2×C4⋊Q8 — C42.414D4
C1C22C2×C4 — C42.414D4
C1C22C2×C42 — C42.414D4
C1C22C22C42 — C42.414D4

Generators and relations for C42.414D4
 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=bc3 >

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, C42.414D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C4.10D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C8.C22 [×2], C2×C4.10D4, C2×D4⋊C4, C23.38D4, C42.414D4

Smallest permutation representation of C42.414D4
On 64 points
Generators in S64
(1 9 23 59)(2 60 24 10)(3 11 17 61)(4 62 18 12)(5 13 19 63)(6 64 20 14)(7 15 21 57)(8 58 22 16)(25 49 46 34)(26 35 47 50)(27 51 48 36)(28 37 41 52)(29 53 42 38)(30 39 43 54)(31 55 44 40)(32 33 45 56)
(1 15 19 61)(2 62 20 16)(3 9 21 63)(4 64 22 10)(5 11 23 57)(6 58 24 12)(7 13 17 59)(8 60 18 14)(25 51 42 40)(26 33 43 52)(27 53 44 34)(28 35 45 54)(29 55 46 36)(30 37 47 56)(31 49 48 38)(32 39 41 50)
(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 15 38 19 31 61 49)(2 52 62 26 20 33 16 43)(3 46 9 36 21 29 63 55)(4 50 64 32 22 39 10 41)(5 44 11 34 23 27 57 53)(6 56 58 30 24 37 12 47)(7 42 13 40 17 25 59 51)(8 54 60 28 18 35 14 45)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222244
type++++++++--
imageC1C2C2C2C2C4C4D4D4D8SD16C4.10D4C8.C22
kernelC42.414D4C4.10D8C4⋊M4(2)C42.12C4C2×C4⋊Q8C2×C4⋊C4C4⋊Q8C42C22×C4C2×C4C2×C4C4C4
# reps1411144224422

Matrix representation of C42.414D4 in GL6(𝔽17)

0160000
100000
001000
000100
000010
000001
,
0160000
100000
0016200
0016100
001616162
0071161
,
5120000
12120000
0014111613
001613141
0016037
00811134
,
12120000
5120000
001211150
0047015
0011056
0013121310

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,16,7,0,0,2,1,16,1,0,0,0,0,16,16,0,0,0,0,2,1],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,14,16,16,8,0,0,11,13,0,11,0,0,16,14,3,13,0,0,13,1,7,4],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,12,4,1,13,0,0,11,7,10,12,0,0,15,0,5,13,0,0,0,15,6,10] >;

C42.414D4 in GAP, Magma, Sage, TeX

C_4^2._{414}D_4
% in TeX

G:=Group("C4^2.414D4");
// GroupNames label

G:=SmallGroup(128,278);
// by ID

G=gap.SmallGroup(128,278);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,1123,1018,248,1971,242]);
// 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=b*c^3>;
// generators/relations

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