Copied to
clipboard

G = C42.219D4order 128 = 27

201st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.219D4, C42.333C23, C4⋊C89C22, (C2×D4)⋊22Q8, D4.3(C2×Q8), D42Q81C2, D4.Q812C2, C4⋊Q856C22, C4.Q86C22, D4⋊Q818C2, C4⋊C4.40C23, (C2×C8).24C23, C2.D817C22, C4.28(C22×Q8), C4⋊M4(2)⋊7C2, (C2×C4).275C24, C23.657(C2×D4), (C22×C4).799D4, C4.101(C8⋊C22), (C2×D4).394C23, (C4×D4).316C22, C4.106(C22⋊Q8), C42.C230C22, M4(2)⋊C416C2, D4⋊C4.23C22, (C2×C42).821C22, (C22×C4).994C23, C23.37D4.2C2, C22.535(C22×D4), C22.45(C22⋊Q8), C2.18(D8⋊C22), C23.37C233C2, (C22×D4).571C22, (C2×M4(2)).64C22, C42⋊C2.116C22, (C2×C4×D4).83C2, C4.85(C2×C4○D4), (C2×C4).99(C2×Q8), C2.23(C2×C8⋊C22), C2.56(C2×C22⋊Q8), (C2×C4).1436(C2×D4), (C2×C4).292(C4○D4), (C2×C4⋊C4).922C22, SmallGroup(128,1809)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.219D4
C1C2C4C2×C4C22×C4C22×D4C2×C4×D4 — C42.219D4
C1C2C2×C4 — C42.219D4
C1C22C2×C42 — C42.219D4
C1C2C2C2×C4 — C42.219D4

Generators and relations for C42.219D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b2c3 >

Subgroups: 428 in 220 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×19], D4 [×4], D4 [×6], Q8 [×4], C23, C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×9], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×2], C24, D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C23×C4, C22×D4, C23.37D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], D4⋊Q8 [×2], D42Q8 [×2], D4.Q8 [×4], C2×C4×D4, C23.37C23, C42.219D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C8⋊C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8⋊C22, D8⋊C22, C42.219D4

Smallest permutation representation of C42.219D4
On 32 points
Generators in S32
(1 19 31 13)(2 14 32 20)(3 21 25 15)(4 16 26 22)(5 23 27 9)(6 10 28 24)(7 17 29 11)(8 12 30 18)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)
(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)
(1 8 31 30)(2 29 32 7)(3 6 25 28)(4 27 26 5)(9 16 23 22)(10 21 24 15)(11 14 17 20)(12 19 18 13)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4N4O4P4Q4R8A8B8C8D
order12222222224···44···444448888
size11112244442···24···488888888

32 irreducible representations

dim111111111222244
type+++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4D4Q8C4○D4C8⋊C22D8⋊C22
kernelC42.219D4C23.37D4C4⋊M4(2)M4(2)⋊C4D4⋊Q8D42Q8D4.Q8C2×C4×D4C23.37C23C42C22×C4C2×D4C2×C4C4C2
# reps121222411224422

Matrix representation of C42.219D4 in GL6(𝔽17)

16130000
910000
0016000
0001600
0000160
0000016
,
100000
010000
0011500
0011600
000101
00161160
,
840000
590000
00160150
001601616
0011610
001010
,
9130000
1280000
00160150
0000161
000010
000110

G:=sub<GL(6,GF(17))| [16,9,0,0,0,0,13,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[8,5,0,0,0,0,4,9,0,0,0,0,0,0,16,16,1,1,0,0,0,0,16,0,0,0,15,16,1,1,0,0,0,16,0,0],[9,12,0,0,0,0,13,8,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0] >;

C42.219D4 in GAP, Magma, Sage, TeX

C_4^2._{219}D_4
% in TeX

G:=Group("C4^2.219D4");
// GroupNames label

G:=SmallGroup(128,1809);
// by ID

G=gap.SmallGroup(128,1809);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,2019,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽