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G = C42.360D4order 128 = 27

56th non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.360D4, C42.714C23, (C2×C8)⋊31D4, C41(C4○D8), C4(C85D4), C4(C84D4), C8.53(C2×D4), C84D424C2, C85D430C2, C4(C4⋊Q16), C4.5(C22×D4), C4⋊Q1625C2, C4(C8.12D4), C8.12D427C2, C4.55(C41D4), (C4×C8).416C22, (C2×C8).594C23, (C2×C4).345C24, (C22×C4).614D4, C23.389(C2×D4), C4⋊Q8.279C22, (C2×Q8).99C23, (C2×D4).111C23, (C2×D8).130C22, C22.2(C41D4), C41D4.151C22, (C22×C8).558C22, C22.26C248C2, (C2×Q16).126C22, C22.605(C22×D4), (C22×C4).1560C23, (C2×C42).1129C22, (C2×SD16).148C22, C4.4D4.139C22, (C2×C4×C8)⋊30C2, (C2×C4○D8)⋊8C2, (C2×C4)(C85D4), (C2×C4)(C84D4), C2.31(C2×C4○D8), (C2×C4)(C4⋊Q16), (C2×C4).855(C2×D4), C2.24(C2×C41D4), (C2×C4)(C8.12D4), (C2×C4○D4).153C22, SmallGroup(128,1879)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.360D4
C1C2C22C2×C4C22×C4C2×C42C2×C4×C8 — C42.360D4
C1C2C2×C4 — C42.360D4
C1C2×C4C2×C42 — C42.360D4
C1C2C2C2×C4 — C42.360D4

Generators and relations for C42.360D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 564 in 286 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×24], Q8 [×8], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×4], C4○D4 [×16], C4×C8 [×2], C4×C8 [×2], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C4○D8 [×16], C2×C4○D4 [×4], C2×C4×C8, C85D4 [×2], C84D4, C4⋊Q16, C8.12D4 [×4], C22.26C24 [×2], C2×C4○D8 [×4], C42.360D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C4○D8 [×4], C22×D4 [×3], C2×C41D4, C2×C4○D8 [×2], C42.360D4

Smallest permutation representation of C42.360D4
On 64 points
Generators in S64
(1 63 55 16)(2 64 56 9)(3 57 49 10)(4 58 50 11)(5 59 51 12)(6 60 52 13)(7 61 53 14)(8 62 54 15)(17 37 31 45)(18 38 32 46)(19 39 25 47)(20 40 26 48)(21 33 27 41)(22 34 28 42)(23 35 29 43)(24 36 30 44)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)
(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 47 5 43)(2 42 6 46)(3 45 7 41)(4 48 8 44)(9 22 13 18)(10 17 14 21)(11 20 15 24)(12 23 16 19)(25 59 29 63)(26 62 30 58)(27 57 31 61)(28 60 32 64)(33 49 37 53)(34 52 38 56)(35 55 39 51)(36 50 40 54)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R8A···8P
order122222222244444···444448···8
size111122888811112···288882···2

44 irreducible representations

dim111111112222
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4C4○D8
kernelC42.360D4C2×C4×C8C85D4C84D4C4⋊Q16C8.12D4C22.26C24C2×C4○D8C42C2×C8C22×C4C4
# reps1121142428216

Matrix representation of C42.360D4 in GL4(𝔽17) generated by

01600
1000
001615
0011
,
13000
01300
0010
0001
,
51200
5500
00160
00016
,
13000
0400
001615
0001
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,1,0,0,15,1],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[5,5,0,0,12,5,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,15,1] >;

C42.360D4 in GAP, Magma, Sage, TeX

C_4^2._{360}D_4
% in TeX

G:=Group("C4^2.360D4");
// GroupNames label

G:=SmallGroup(128,1879);
// by ID

G=gap.SmallGroup(128,1879);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,248,2804,172]);
// Polycyclic

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

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