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G = M4(2).43D4order 128 = 27

7th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).43D4, (C2×D8)⋊5C4, (C2×Q16)⋊5C4, C4.80(C4×D4), C4○D4.24D4, (C2×C8).321D4, (C2×SD16)⋊13C4, C2.14(C8○D8), C4.124C22≀C2, C4.C423C2, D4.1(C22⋊C4), C22.150(C4×D4), Q8.1(C22⋊C4), C2.14(C8.26D4), C4.187(C4⋊D4), C22.1(C4⋊D4), C23.204(C4○D4), (C22×C8).388C22, (C2×C42).276C22, (C22×C4).1363C23, C22.7C4224C2, C2.15(C23.23D4), (C2×M4(2)).177C22, C22.20(C22.D4), (C2×C4≀C2)⋊13C2, (C2×C8○D4)⋊14C2, (C2×C4○D8).1C2, (C2×C8).108(C2×C4), (C2×D4).72(C2×C4), (C2×C4).995(C2×D4), C4.10(C2×C22⋊C4), (C2×Q8).63(C2×C4), (C22×C8)⋊C222C2, (C2×C4).865(C4○D4), (C2×C4).381(C22×C4), (C2×C4○D4).263C22, SmallGroup(128,608)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).43D4
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — M4(2).43D4
C1C2C2×C4 — M4(2).43D4
C1C2×C4C22×C8 — M4(2).43D4
C1C2C2C22×C4 — M4(2).43D4

Generators and relations for M4(2).43D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=a-1b, dad-1=a3b, cbc-1=a4b, bd=db, dcd-1=a4bc3 >

Subgroups: 308 in 162 conjugacy classes, 56 normal (46 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×5], C22 [×3], C22 [×9], C8 [×7], C2×C4 [×6], C2×C4 [×12], D4 [×2], D4 [×9], Q8 [×2], Q8 [×3], C23, C23 [×2], C42 [×2], C2×C8 [×4], C2×C8 [×10], M4(2) [×2], M4(2) [×7], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C22⋊C8 [×2], C4≀C2 [×4], C2×C42, C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4 [×2], C22.7C42, C4.C42, (C22×C8)⋊C2, C2×C4≀C2 [×2], C2×C8○D4, C2×C4○D8, M4(2).43D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C8○D8, C8.26D4, M4(2).43D4

Smallest permutation representation of M4(2).43D4
On 32 points
Generators in S32
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 9 27 20 5 13 31 24)(2 16 28 19 6 12 32 23)(3 11 29 22 7 15 25 18)(4 10 30 21 8 14 26 17)
(2 4 6 8)(9 20 13 24)(10 19)(11 22 15 18)(12 21)(14 23)(16 17)(26 28 30 32)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G···4L4M8A8B8C8D8E···8N8O8P
order1222222224444444···4488888···888
size1111224481111224···4822224···488

38 irreducible representations

dim11111111112222224
type++++++++++
imageC1C2C2C2C2C2C2C4C4C4D4D4D4C4○D4C4○D4C8○D8C8.26D4
kernelM4(2).43D4C22.7C42C4.C42(C22×C8)⋊C2C2×C4≀C2C2×C8○D4C2×C4○D8C2×D8C2×SD16C2×Q16C2×C8M4(2)C4○D4C2×C4C23C2C2
# reps11112112422242282

Matrix representation of M4(2).43D4 in GL4(𝔽17) generated by

0100
16000
00213
00215
,
1000
0100
0010
00116
,
16000
0100
00138
0064
,
1000
01600
0010
001113
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,2,2,0,0,13,15],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,13,6,0,0,8,4],[1,0,0,0,0,16,0,0,0,0,1,11,0,0,0,13] >;

M4(2).43D4 in GAP, Magma, Sage, TeX

M_4(2)._{43}D_4
% in TeX

G:=Group("M4(2).43D4");
// GroupNames label

G:=SmallGroup(128,608);
// by ID

G=gap.SmallGroup(128,608);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,1411,718,172,2028,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*b*c^3>;
// generators/relations

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