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G = C42.5Q8order 128 = 27

5th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.5Q8, C42.22D4, C4⋊C82C4, (C4×C8)⋊2C4, (C2×C4).8D8, (C2×C4).4Q16, C4.7(C4.Q8), C4.8(C2.D8), (C2×C4).30C42, (C2×C4).11SD16, C428C4.1C2, C42.304(C2×C4), (C22×C4).176D4, C4.28(D4⋊C4), C2.5(C4.9C42), C4.20(Q8⋊C4), C4⋊M4(2).4C2, C2.C42.1C4, C42.12C4.7C2, C2.3(C22.4Q16), (C2×C42).126C22, C22.10(D4⋊C4), C2.4(M4(2)⋊4C4), C23.138(C22⋊C4), C22.13(Q8⋊C4), C22.36(C2.C42), (C2×C4).95(C4⋊C4), (C22×C4).146(C2×C4), (C2×C4).336(C22⋊C4), SmallGroup(128,18)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.5Q8
C1C2C22C2×C4C22×C4C2×C42C42.12C4 — C42.5Q8
C1C2C2×C4 — C42.5Q8
C1C22C2×C42 — C42.5Q8
C1C22C22C2×C42 — C42.5Q8

Generators and relations for C42.5Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a-1b2c3 >

Subgroups: 160 in 78 conjugacy classes, 40 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×10], C2×C4 [×8], C23, C42 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C2.C42, C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C4⋊C4, C2×M4(2), C428C4, C4⋊M4(2), C42.12C4, C42.5Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C4.9C42, C22.4Q16, M4(2)⋊4C4, C42.5Q8

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim111111122222244
type+++++-++-
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16C4.9C42M4(2)⋊4C4
kernelC42.5Q8C428C4C4⋊M4(2)C42.12C4C2.C42C4×C8C4⋊C8C42C42C22×C4C2×C4C2×C4C2×C4C2C2
# reps111144411224222

Matrix representation of C42.5Q8 in GL6(𝔽17)

1600000
0160000
0016900
0013100
0000169
0000131
,
4150000
0130000
0013200
001400
00312415
00601613
,
870000
020000
0003150
001012015
0000014
000075
,
1300000
140000
001000
0041600
006140
0015111613

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,9,1,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,13,1,3,6,0,0,2,4,12,0,0,0,0,0,4,16,0,0,0,0,15,13],[8,0,0,0,0,0,7,2,0,0,0,0,0,0,0,10,0,0,0,0,3,12,0,0,0,0,15,0,0,7,0,0,0,15,14,5],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,6,15,0,0,0,16,1,11,0,0,0,0,4,16,0,0,0,0,0,13] >;

C42.5Q8 in GAP, Magma, Sage, TeX

C_4^2._5Q_8
% in TeX

G:=Group("C4^2.5Q8");
// GroupNames label

G:=SmallGroup(128,18);
// by ID

G=gap.SmallGroup(128,18);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,102]);
// Polycyclic

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

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