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G = C42.6Q8order 128 = 27

6th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.6Q8, C42.24D4, (C4×C8)⋊7C4, C8⋊C44C4, C22.10C4≀C2, (C2×C4).32C42, C425C4.1C2, C42.290(C2×C4), (C22×C4).177D4, (C4×M4(2)).9C2, C2.7(C4.9C42), C2.6(C426C4), C2.C42.2C4, C42.6C4.6C2, (C2×C42).128C22, C2.6(M4(2)⋊4C4), C23.139(C22⋊C4), C22.38(C2.C42), (C2×C4).15(C4⋊C4), (C22×C4).147(C2×C4), (C2×C4).295(C22⋊C4), SmallGroup(128,20)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.6Q8
C1C2C22C2×C4C22×C4C2×C42C4×M4(2) — C42.6Q8
C1C2C2×C4 — C42.6Q8
C1C22C2×C42 — C42.6Q8
C1C22C22C2×C42 — C42.6Q8

Generators and relations for C42.6Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=ab2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=ac3 >

Subgroups: 152 in 76 conjugacy classes, 32 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×10], C23, C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C2.C42 [×2], C4×C8 [×2], C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×M4(2), C425C4, C4×M4(2), C42.6C4, C42.6Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×2], C4.9C42, C426C4, M4(2)⋊4C4, C42.6Q8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222244
type+++++-+
imageC1C2C2C2C4C4C4D4Q8D4C4≀C2C4.9C42M4(2)⋊4C4
kernelC42.6Q8C425C4C4×M4(2)C42.6C4C2.C42C4×C8C8⋊C4C42C42C22×C4C22C2C2
# reps1111444112822

Matrix representation of C42.6Q8 in GL6(𝔽17)

1300000
0130000
001900
0001600
000314
0000016
,
010000
100000
004200
0001300
001412131
000004
,
10110000
11100000
00120150
000001
0021650
0001300
,
400000
0130000
001000
00131600
0016040
001201513

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

C42.6Q8 in GAP, Magma, Sage, TeX

C_4^2._6Q_8
% in TeX

G:=Group("C4^2.6Q8");
// GroupNames label

G:=SmallGroup(128,20);
// by ID

G=gap.SmallGroup(128,20);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,172]);
// 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*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*c^3>;
// generators/relations

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