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G = Q8.C42order 128 = 27

2nd non-split extension by Q8 of C42 acting via C42/C2×C4=C2

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

Aliases: D4.2C42, Q8.2C42, C82C4≀C2, C4≀C26C4, C8○D46C4, C2.4(C8○D8), C4.163(C4×D4), (C2×C8).384D4, C4.7(C2×C42), C83(D4⋊C4), D4⋊C413C4, C8(C426C4), C83(Q8⋊C4), Q8⋊C413C4, C22.28(C4×D4), C426C435C2, C8.61(C22⋊C4), C8(C4.C42), C42.262(C2×C4), C82M4(2)⋊22C2, C4.C4223C2, C8(C23.24D4), M4(2).17(C2×C4), C23.199(C4○D4), (C22×C8).474C22, (C22×C4).1311C23, (C2×C42).1048C22, C22.1(C42⋊C2), C23.24D4.13C2, C42⋊C2.263C22, (C2×M4(2)).308C22, C8(C2×C4≀C2), (C2×C8)C4≀C2, (C2×C4×C8)⋊38C2, (C2×C4≀C2).17C2, (C2×C8)(D4⋊C4), C4⋊C4.142(C2×C4), (C2×C8).191(C2×C4), (C2×C8)(Q8⋊C4), C4○D4.25(C2×C4), (C2×C8○D4).12C2, C2.22(C4×C22⋊C4), (C2×D4).159(C2×C4), (C2×C4).1306(C2×D4), C4.110(C2×C22⋊C4), (C2×Q8).141(C2×C4), (C2×C4).542(C4○D4), (C2×C8)(C4.C42), (C2×C4).354(C22×C4), (C2×C4○D4).252C22, (C2×C8)(C23.24D4), SmallGroup(128,496)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q8.C42
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — Q8.C42
C1C2C4 — Q8.C42
C1C2×C8C22×C8 — Q8.C42
C1C2C2C22×C4 — Q8.C42

Generators and relations for Q8.C42
 G = < a,b,c,d | a4=c4=1, b2=d4=a2, bab-1=dad-1=a-1, dcd-1=ac=ca, cbc-1=a-1b, bd=db >

Subgroups: 236 in 150 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×13], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×8], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4×C8 [×3], C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2, C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, C426C4, C4.C42, C2×C4×C8, C82M4(2), C23.24D4, C2×C4≀C2, C2×C8○D4, Q8.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C8○D8 [×2], Q8.C42

Smallest permutation representation of Q8.C42
On 32 points
Generators in S32
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 18 13 22)(10 23 14 19)(11 20 15 24)(12 17 16 21)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)
(1 7 5 3)(2 26)(4 28)(6 30)(8 32)(9 20)(10 16 14 12)(11 22)(13 24)(15 18)(17 23 21 19)(25 31 29 27)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8H8I···8T8U···8AB
order1222222244444···44···48···88···88···8
size1111224411112···24···41···12···24···4

56 irreducible representations

dim1111111111112222
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4C4○D4C8○D8
kernelQ8.C42C426C4C4.C42C2×C4×C8C82M4(2)C23.24D4C2×C4≀C2C2×C8○D4D4⋊C4Q8⋊C4C4≀C2C8○D4C2×C8C2×C4C23C2
# reps11111111448842216

Matrix representation of Q8.C42 in GL4(𝔽17) generated by

16000
01600
0040
001313
,
0100
1000
00816
00149
,
1000
01600
0040
00616
,
0100
1000
0012
00616
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,13,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,8,14,0,0,16,9],[1,0,0,0,0,16,0,0,0,0,4,6,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,1,6,0,0,2,16] >;

Q8.C42 in GAP, Magma, Sage, TeX

Q_8.C_4^2
% in TeX

G:=Group("Q8.C4^2");
// GroupNames label

G:=SmallGroup(128,496);
// by ID

G=gap.SmallGroup(128,496);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,172,4037]);
// Polycyclic

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

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