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G = C42.62Q8order 128 = 27

22nd non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Aliases: C42.62Q8, C2.D88C4, C4.Q813C4, C8.21(C4⋊C4), (C2×C8).50Q8, C4.10(C4×Q8), (C2×C8).228D4, C4.50(C4⋊Q8), C2.18(C8○D8), C426C4.8C2, C22.178(C4×D4), C4.199(C4⋊D4), C23.209(C4○D4), (C22×C8).488C22, C23.25D4.3C2, C22.25(C22⋊Q8), C22.3(C42.C2), C42⋊C2.40C22, (C2×C42).1069C22, (C22×C4).1394C23, (C2×M4(2)).202C22, C42.6C22.11C2, C2.12(C23.65C23), (C2×C4×C8).46C2, C4.43(C2×C4⋊C4), C4⋊C4.91(C2×C4), (C2×C8).184(C2×C4), (C2×C4).206(C2×Q8), (C2×C4).1540(C2×D4), (C2×C8.C4).13C2, (C2×C4).589(C4○D4), (C2×C4).412(C22×C4), SmallGroup(128,677)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.62Q8
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C42.62Q8
C1C2C2×C4 — C42.62Q8
C1C2×C4C22×C8 — C42.62Q8
C1C2C2C22×C4 — C42.62Q8

Generators and relations for C42.62Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b2c2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=bc3 >

Subgroups: 164 in 104 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×2], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4, C4×C8 [×2], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C8.C4 [×2], C2×C42, C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C426C4 [×2], C2×C4×C8, C42.6C22 [×2], C23.25D4, C2×C8.C4, C42.62Q8
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], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C8○D8 [×2], C42.62Q8

Smallest permutation representation of C42.62Q8
On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 11 13 15)(10 12 14 16)(25 27 29 31)(26 28 30 32)
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 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 28 7 26 5 32 3 30)(2 14 8 12 6 10 4 16)(9 24 15 22 13 20 11 18)(17 31 23 29 21 27 19 25)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O4P4Q4R8A···8P8Q8R8S8T
order12222244444···444448···88888
size11112211112···288882···28888

44 irreducible representations

dim11111111222222
type++++++-+-
imageC1C2C2C2C2C2C4C4Q8D4Q8C4○D4C4○D4C8○D8
kernelC42.62Q8C426C4C2×C4×C8C42.6C22C23.25D4C2×C8.C4C4.Q8C2.D8C42C2×C8C2×C8C2×C4C23C2
# reps121211442422216

Matrix representation of C42.62Q8 in GL4(𝔽17) generated by

16600
0400
0010
001113
,
4000
0400
0040
0004
,
2200
01500
0080
0008
,
13600
8400
00115
00716
G:=sub<GL(4,GF(17))| [16,0,0,0,6,4,0,0,0,0,1,11,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,2,15,0,0,0,0,8,0,0,0,0,8],[13,8,0,0,6,4,0,0,0,0,1,7,0,0,15,16] >;

C42.62Q8 in GAP, Magma, Sage, TeX

C_4^2._{62}Q_8
% in TeX

G:=Group("C4^2.62Q8");
// GroupNames label

G:=SmallGroup(128,677);
// by ID

G=gap.SmallGroup(128,677);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2804,172,124]);
// Polycyclic

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

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