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G = C42.28Q8order 128 = 27

28th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.28Q8, C4.Q88C4, C8.6(C4⋊C4), C2.D813C4, (C2×C8).14Q8, C4.11(C4×Q8), (C2×C8).203D4, C4.51(C4⋊Q8), C426C4.9C2, C22.179(C4×D4), C2.18(C8.26D4), C4.200(C4⋊D4), C23.210(C4○D4), (C22×C8).402C22, (C2×C42).309C22, C22.26(C22⋊Q8), C22.4(C42.C2), C42⋊C2.41C22, (C22×C4).1395C23, C23.25D4.13C2, (C2×M4(2)).203C22, C42.6C22.12C2, C2.13(C23.65C23), C4.44(C2×C4⋊C4), C4⋊C4.92(C2×C4), (C2×C8).67(C2×C4), (C2×C8⋊C4).8C2, (C2×C4).207(C2×Q8), (C2×C4).1541(C2×D4), (C2×C8.C4).14C2, (C2×C4).590(C4○D4), (C2×C4).413(C22×C4), SmallGroup(128,678)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.28Q8
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C42.28Q8
C1C2C2×C4 — C42.28Q8
C1C2×C4C22×C8 — C42.28Q8
C1C2C2C22×C4 — C42.28Q8

Generators and relations for C42.28Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b2c2, ab=ba, cac-1=ab2, dad-1=a-1b, bc=cb, bd=db, dcd-1=bc3 >

Subgroups: 164 in 100 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×8], 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, C8⋊C4 [×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×C8⋊C4, C42.6C22 [×2], C23.25D4, C2×C8.C4, C42.28Q8
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, C8.26D4 [×2], C42.28Q8

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A···8H8I8J8K8L
order122222444444444444448···88888
size111122111122444488884···48888

32 irreducible representations

dim11111111222224
type++++++-+-
imageC1C2C2C2C2C2C4C4Q8D4Q8C4○D4C4○D4C8.26D4
kernelC42.28Q8C426C4C2×C8⋊C4C42.6C22C23.25D4C2×C8.C4C4.Q8C2.D8C42C2×C8C2×C8C2×C4C23C2
# reps12121144242224

Matrix representation of C42.28Q8 in GL6(𝔽17)

400000
040000
001006
00016110
000040
0000013
,
1600000
0160000
004000
000400
000040
000004
,
040000
1300000
00016110
004006
0000016
000040
,
0160000
1600000
00013100
00130010
009004
000940

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

C42.28Q8 in GAP, Magma, Sage, TeX

C_4^2._{28}Q_8
% in TeX

G:=Group("C4^2.28Q8");
// GroupNames label

G:=SmallGroup(128,678);
// by ID

G=gap.SmallGroup(128,678);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,100,2804,172,124]);
// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

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