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G = C42.446D4order 128 = 27

79th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.446D4, C42.325C23, C4⋊C84C22, D4.6(C2×D4), Q8.6(C2×D4), C4○D4.30D4, (C4×D4)⋊84C22, (C2×C8).15C23, (C4×Q8)⋊80C22, C4.71(C22×D4), D4.2D413C2, C4⋊C4.381C23, C4⋊M4(2)⋊6C2, (C2×C4).244C24, Q8.D413C2, (C2×Q16)⋊16C22, (C2×SD16)⋊8C22, (C2×D8).54C22, C23.656(C2×D4), (C22×C4).424D4, (C2×Q8).38C23, C4.106(C4⋊D4), Q8⋊C418C22, (C2×D4).387C23, C23.38D47C2, C23.37D47C2, D4⋊C4.21C22, C22.79(C4⋊D4), (C2×C42).813C22, (C22×C4).974C23, C22.504(C22×D4), C2.13(D8⋊C22), C4.4D4.127C22, (C22×D4).339C22, (C2×M4(2)).51C22, (C22×Q8).272C22, C42⋊C2.313C22, (C4×C4○D4)⋊8C2, C4.154(C2×C4○D4), C2.62(C2×C4⋊D4), (C2×C4.4D4)⋊39C2, (C2×C4).1214(C2×D4), (C2×C8.C22)⋊15C2, (C2×C8⋊C22).10C2, (C2×C4).275(C4○D4), (C2×C4○D4).300C22, SmallGroup(128,1772)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.446D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.446D4
C1C2C2×C4 — C42.446D4
C1C22C2×C42 — C42.446D4
C1C2C2C2×C4 — C42.446D4

Generators and relations for C42.446D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1b2, cbc-1=dbd=b-1, dcd=b2c3 >

Subgroups: 500 in 250 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], D4 [×2], D4 [×11], Q8 [×2], Q8 [×7], C23, C23 [×9], C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.37D4, C23.38D4, C4⋊M4(2), D4.2D4 [×4], Q8.D4 [×4], C4×C4○D4, C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.446D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D8⋊C22 [×2], C42.446D4

Smallest permutation representation of C42.446D4
On 32 points
Generators in S32
(1 30 19 14)(2 11 20 27)(3 32 21 16)(4 13 22 29)(5 26 23 10)(6 15 24 31)(7 28 17 12)(8 9 18 25)
(1 17 5 21)(2 22 6 18)(3 19 7 23)(4 24 8 20)(9 27 13 31)(10 32 14 28)(11 29 15 25)(12 26 16 30)
(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)
(2 8)(3 7)(4 6)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 32)(17 21)(18 20)(22 24)

G:=sub<Sym(32)| (1,30,19,14)(2,11,20,27)(3,32,21,16)(4,13,22,29)(5,26,23,10)(6,15,24,31)(7,28,17,12)(8,9,18,25), (1,17,5,21)(2,22,6,18)(3,19,7,23)(4,24,8,20)(9,27,13,31)(10,32,14,28)(11,29,15,25)(12,26,16,30), (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), (2,8)(3,7)(4,6)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,32)(17,21)(18,20)(22,24)>;

G:=Group( (1,30,19,14)(2,11,20,27)(3,32,21,16)(4,13,22,29)(5,26,23,10)(6,15,24,31)(7,28,17,12)(8,9,18,25), (1,17,5,21)(2,22,6,18)(3,19,7,23)(4,24,8,20)(9,27,13,31)(10,32,14,28)(11,29,15,25)(12,26,16,30), (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), (2,8)(3,7)(4,6)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,32)(17,21)(18,20)(22,24) );

G=PermutationGroup([(1,30,19,14),(2,11,20,27),(3,32,21,16),(4,13,22,29),(5,26,23,10),(6,15,24,31),(7,28,17,12),(8,9,18,25)], [(1,17,5,21),(2,22,6,18),(3,19,7,23),(4,24,8,20),(9,27,13,31),(10,32,14,28),(11,29,15,25),(12,26,16,30)], [(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)], [(2,8),(3,7),(4,6),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,32),(17,21),(18,20),(22,24)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4P4Q4R8A8B8C8D
order12222222224···44···4448888
size11112244882···24···4888888

32 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D8⋊C22
kernelC42.446D4C23.37D4C23.38D4C4⋊M4(2)D4.2D4Q8.D4C4×C4○D4C2×C4.4D4C2×C8⋊C22C2×C8.C22C42C22×C4C4○D4C2×C4C2
# reps111144111122444

Matrix representation of C42.446D4 in GL6(𝔽17)

040000
400000
004900
0041300
0001304
00413130
,
1600000
0160000
0011500
0011600
0001601
00116160
,
0160000
100000
0010015
0000116
0000016
00116016
,
100000
0160000
001000
0011600
000001
000010

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,4,0,0,9,13,13,13,0,0,0,0,0,13,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,1,0,0,0,0,15,16,16,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C42.446D4 in GAP, Magma, Sage, TeX

C_4^2._{446}D_4
% in TeX

G:=Group("C4^2.446D4");
// GroupNames label

G:=SmallGroup(128,1772);
// by ID

G=gap.SmallGroup(128,1772);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2019,248,4037,1027,124]);
// Polycyclic

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

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