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G = C42.430D4order 128 = 27

63rd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.430D4, M4(2).4Q8, M4(2).26D4, C4⋊C8.16C4, C4.38(C4×Q8), C4.146(C4×D4), C42(C8.C4), C22.6(C4⋊Q8), C23.86(C2×Q8), (C22×C4).52Q8, C42.153(C2×C4), C4.201(C4⋊D4), (C4×M4(2)).24C2, C4.C42.3C2, C4.119(C22⋊Q8), C4⋊M4(2).31C2, (C2×C42).313C22, (C22×C8).167C22, C22.8(C42.C2), (C22×C4).1399C23, C2.12(M4(2).C4), (C2×M4(2)).204C22, C2.17(C23.65C23), (C2×C4⋊C8).45C2, (C2×C8).45(C2×C4), (C2×C4).56(C4⋊C4), (C2×C4).274(C2×Q8), (C2×C4).1542(C2×D4), C2.15(C2×C8.C4), C22.117(C2×C4⋊C4), (C2×C8.C4).15C2, (C2×C4).763(C4○D4), (C2×C4).555(C22×C4), SmallGroup(128,682)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.430D4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — C42.430D4
C1C2C2×C4 — C42.430D4
C1C2×C4C2×C42 — C42.430D4
C1C2C2C22×C4 — C42.430D4

Generators and relations for C42.430D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 148 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C4 [×3], C22 [×3], C22 [×2], C8 [×10], C2×C4 [×10], C2×C4 [×3], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×4], M4(2) [×6], C22×C4 [×3], C4×C8, C8⋊C4, C4⋊C8 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C8.C4 [×4], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C4.C42 [×2], C4×M4(2), C2×C4⋊C8, C4⋊M4(2), C2×C8.C4 [×2], C42.430D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C8.C4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C2×C8.C4, M4(2).C4, C42.430D4

Smallest permutation representation of C42.430D4
On 64 points
Generators in S64
(1 54 59 21)(2 22 60 55)(3 56 61 23)(4 24 62 49)(5 50 63 17)(6 18 64 51)(7 52 57 19)(8 20 58 53)(9 41 36 29)(10 30 37 42)(11 43 38 31)(12 32 39 44)(13 45 40 25)(14 26 33 46)(15 47 34 27)(16 28 35 48)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(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 28 7 30 5 32 3 26)(2 31 8 25 6 27 4 29)(9 55 11 53 13 51 15 49)(10 50 12 56 14 54 16 52)(17 39 23 33 21 35 19 37)(18 34 24 36 22 38 20 40)(41 60 43 58 45 64 47 62)(42 63 44 61 46 59 48 57)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K4L8A···8P8Q8R8S8T
order12222244444···4448···88888
size11112211112···2444···48888

38 irreducible representations

dim11111112222224
type++++++++--
imageC1C2C2C2C2C2C4D4D4Q8Q8C4○D4C8.C4M4(2).C4
kernelC42.430D4C4.C42C4×M4(2)C2×C4⋊C8C4⋊M4(2)C2×C8.C4C4⋊C8C42M4(2)M4(2)C22×C4C2×C4C4C2
# reps12111282222482

Matrix representation of C42.430D4 in GL4(𝔽17) generated by

1000
0100
0001
00160
,
4000
0400
0010
0001
,
9000
01500
00411
001113
,
0800
9000
00136
0064
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,15,0,0,0,0,4,11,0,0,11,13],[0,9,0,0,8,0,0,0,0,0,13,6,0,0,6,4] >;

C42.430D4 in GAP, Magma, Sage, TeX

C_4^2._{430}D_4
% in TeX

G:=Group("C4^2.430D4");
// GroupNames label

G:=SmallGroup(128,682);
// by ID

G=gap.SmallGroup(128,682);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=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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