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G = C2×D4.8D4order 128 = 27

Direct product of C2 and D4.8D4

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

Aliases: C2×D4.8D4, C42.311C23, M4(2).1C23, 2- 1+45C22, C4≀C29C22, C4○D4.16D4, D4.50(C2×D4), (C2×C4).8C24, Q8.50(C2×D4), C4.45C22≀C2, (C2×D4).298D4, C8⋊C225C22, (C2×Q8).233D4, C4○D4.3C23, C4.53(C22×D4), (C2×D4).34C23, C23.650(C2×D4), (C22×C4).108D4, (C2×Q8).25C23, C4.10D46C22, C4.4D450C22, (C2×2- 1+4)⋊3C2, C22.32(C22×D4), C22.120C22≀C2, (C22×C4).967C23, (C2×C42).800C22, (C22×D4).330C22, (C2×M4(2)).44C22, (C22×Q8).264C22, (C2×C4≀C2)⋊6C2, (C2×C4).23(C2×D4), (C2×C8⋊C22)⋊12C2, C2.53(C2×C22≀C2), (C2×C4.4D4)⋊37C2, (C2×C4.10D4)⋊7C2, (C2×C4○D4).106C22, SmallGroup(128,1748)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×D4.8D4
C1C2C22C2×C4C22×C4C22×Q8C2×2- 1+4 — C2×D4.8D4
C1C2C2×C4 — C2×D4.8D4
C1C22C22×C4 — C2×D4.8D4
C1C2C2C2×C4 — C2×D4.8D4

Generators and relations for C2×D4.8D4
 G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >

Subgroups: 708 in 364 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×10], C22 [×3], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×34], D4 [×4], D4 [×24], Q8 [×4], Q8 [×18], C23, C23 [×10], C42 [×2], C42, C22⋊C4 [×8], C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×7], C2×D4 [×4], C2×D4 [×9], C2×Q8 [×6], C2×Q8 [×22], C4○D4 [×8], C4○D4 [×36], C24, C4.10D4 [×4], C4≀C2 [×8], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4, C22×Q8, C22×Q8 [×2], C2×C4○D4 [×2], C2×C4○D4 [×4], 2- 1+4 [×4], 2- 1+4 [×6], C2×C4.10D4, C2×C4≀C2 [×2], D4.8D4 [×8], C2×C4.4D4, C2×C8⋊C22 [×2], C2×2- 1+4, C2×D4.8D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], D4.8D4 [×2], C2×C22≀C2, C2×D4.8D4

Smallest permutation representation of C2×D4.8D4
On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 24 13 20)(10 21 14 17)(11 18 15 22)(12 23 16 19)
(1 16)(2 24)(3 10)(4 18)(5 12)(6 20)(7 14)(8 22)(9 26)(11 28)(13 30)(15 32)(17 31)(19 25)(21 27)(23 29)
(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 27)(2 26)(3 25)(4 32)(5 31)(6 30)(7 29)(8 28)(9 13)(10 12)(14 16)(17 23)(18 22)(19 21)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E···4P8A8B8C8D
order12222222222244444···48888
size11112244448822224···48888

32 irreducible representations

dim111111122224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4D4D4.8D4
kernelC2×D4.8D4C2×C4.10D4C2×C4≀C2D4.8D4C2×C4.4D4C2×C8⋊C22C2×2- 1+4C22×C4C2×D4C2×Q8C4○D4C2
# reps112812122444

Matrix representation of C2×D4.8D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
000100
0016000
008801
0089160
,
1150000
0160000
0022013
0000130
000400
00511515
,
1150000
1160000
000010
0099016
000100
00141398
,
100000
1160000
001000
0001600
008801
009810

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,8,8,0,0,1,0,8,9,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,2,0,0,5,0,0,2,0,4,1,0,0,0,13,0,15,0,0,13,0,0,15],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,9,0,14,0,0,0,9,1,13,0,0,1,0,0,9,0,0,0,16,0,8],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,8,9,0,0,0,16,8,8,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D4.8D4 in GAP, Magma, Sage, TeX

C_2\times D_4._8D_4
% in TeX

G:=Group("C2xD4.8D4");
// GroupNames label

G:=SmallGroup(128,1748);
// by ID

G=gap.SmallGroup(128,1748);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,718,172,2028]);
// Polycyclic

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

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