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G = C42.355D4order 128 = 27

51st non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.355D4, C42.706C23, C4(C4.4D8), C4.23(C4○D8), C4.4D845C2, C4⋊C4.84C23, C4(C4.SD16), (C2×C8).491C23, (C4×C8).383C22, (C2×C4).329C24, C4.SD1646C2, (C2×D4).98C23, C23.387(C2×D4), (C22×C4).610D4, C4⋊Q8.272C22, (C2×Q8).86C23, C4.97(C4.4D4), C23.24D45C2, C41D4.144C22, (C22×C8).520C22, C22.4(C4.4D4), C22.589(C22×D4), D4⋊C4.144C22, C4(C42.78C22), C23.37C238C2, (C22×C4).1551C23, (C2×C42).1124C22, Q8⋊C4.136C22, C4.4D4.133C22, C42.C2.109C22, C42⋊C2.137C22, C42.78C2232C2, C22.26C24.33C2, (C2×C4×C8)⋊22C2, C2.29(C2×C4○D8), C4.38(C2×C4○D4), (C2×C4)(C4.4D8), (C2×C4).694(C2×D4), (C2×C4)(C4.SD16), C2.40(C2×C4.4D4), (C2×C4).708(C4○D4), (C2×C4○D4).147C22, (C2×C4)(C42.78C22), SmallGroup(128,1863)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.355D4
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C42.355D4
C1C2C2×C4 — C42.355D4
C1C2×C4C2×C42 — C42.355D4
C1C2C2C2×C4 — C42.355D4

Generators and relations for C42.355D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >

Subgroups: 372 in 200 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42.C2 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×2], C22×C8 [×2], C2×C4○D4 [×2], C2×C4×C8, C23.24D4 [×4], C4.4D8 [×2], C4.SD16 [×2], C42.78C22 [×4], C22.26C24, C23.37C23, C42.355D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C4○D8 [×2], C42.355D4

Smallest permutation representation of C42.355D4
On 64 points
Generators in S64
(1 60 55 15)(2 61 56 16)(3 62 49 9)(4 63 50 10)(5 64 51 11)(6 57 52 12)(7 58 53 13)(8 59 54 14)(17 40 31 42)(18 33 32 43)(19 34 25 44)(20 35 26 45)(21 36 27 46)(22 37 28 47)(23 38 29 48)(24 39 30 41)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)
(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 50 55 4)(2 3 56 49)(5 54 51 8)(6 7 52 53)(9 16 62 61)(10 60 63 15)(11 14 64 59)(12 58 57 13)(17 30 31 24)(18 23 32 29)(19 28 25 22)(20 21 26 27)(33 48 43 38)(34 37 44 47)(35 46 45 36)(39 42 41 40)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111228811112···28···82···2

44 irreducible representations

dim111111112222
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4C4○D8
kernelC42.355D4C2×C4×C8C23.24D4C4.4D8C4.SD16C42.78C22C22.26C24C23.37C23C42C22×C4C2×C4C4
# reps1142241122816

Matrix representation of C42.355D4 in GL4(𝔽17) generated by

4800
01300
0001
00160
,
1000
0100
00130
00013
,
4000
0400
0055
00125
,
4000
131300
00512
001212
G:=sub<GL(4,GF(17))| [4,0,0,0,8,13,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,5,12,0,0,5,5],[4,13,0,0,0,13,0,0,0,0,5,12,0,0,12,12] >;

C42.355D4 in GAP, Magma, Sage, TeX

C_4^2._{355}D_4
% in TeX

G:=Group("C4^2.355D4");
// GroupNames label

G:=SmallGroup(128,1863);
// by ID

G=gap.SmallGroup(128,1863);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,248,2804,172,4037,124]);
// Polycyclic

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

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