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G = C42.28Q8  order 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.Q8⋊8C4, C8.6(C4⋊C4), C2.D8⋊13C4, (C2×C8).14Q8, C4.11(C4×Q8), (C2×C8).203D4, C4.51(C4⋊Q8), C42⋊6C4.9C2, 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

Derived series
C1 — C2×C4 — C42.28Q8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C42.28Q8
Lower central
C1 — C2 — C2×C4 — C42.28Q8
Upper central
C1 — C2×C4 — C22×C8 — C42.28Q8
Jennings
C1 — C2 — C2 — C22×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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C8⋊C4, C4⋊C8, C4.Q8, C2.D8, C8.C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C42⋊6C4, C2×C8⋊C4, C42.6C22, C23.25D4, C2×C8.C4, C42.28Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C8.26D4, 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 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)

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

32 conjugacy classes

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

32 irreducible representations

dim11111111222224
type++++++-+-
imageC1C2C2C2C2C2C4C4Q8D4Q8C4○D4C4○D4C8.26D4
kernelC42.28Q8C42⋊6C4C2×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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