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G = C8.(C4⋊C4)  order 128 = 27

4th non-split extension by C8 of C4⋊C4 acting via C4⋊C4/C22=C22

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

Aliases: (C4×C8)⋊5C4, C8⋊C48C4, C8.4(C4⋊C4), (C2×C8).9Q8, (C2×C8).107D4, C42.21(C2×C4), C4.9C42.3C2, C23.5(C4○D4), C2.6(C428C4), C4.47(C4.4D4), C82M4(2).1C2, C4.10(C42.C2), C4.86(C42⋊C2), M4(2)⋊4C4.3C2, (C22×C4).670C23, (C22×C8).219C22, C22.6(C42.C2), C23.25D4.12C2, C22.17(C4.4D4), C42⋊C2.269C22, C22.32(C42⋊C2), (C2×M4(2)).166C22, C4.32(C2×C4⋊C4), (C2×C8).13(C2×C4), (C2×C4).237(C2×D4), (C2×C4).115(C2×Q8), (C2×C4).49(C4○D4), (C2×C8.C4).10C2, (C2×C4).535(C22×C4), SmallGroup(128,565)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.(C4⋊C4)
C1C2C22C2×C4C22×C4C42⋊C2C82M4(2) — C8.(C4⋊C4)
C1C2C2×C4 — C8.(C4⋊C4)
C1C4C42⋊C2 — C8.(C4⋊C4)
C1C2C2C22×C4 — C8.(C4⋊C4)

Generators and relations for C8.(C4⋊C4)
 G = < a,b,c | a8=c4=1, b4=a4, bab-1=a5, cac-1=a-1, cbc-1=a6b3 >

Subgroups: 148 in 86 conjugacy classes, 50 normal (34 characteristic)
C1, 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, C4×C8, C8⋊C4, C4.Q8, C2.D8, C8.C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C4.9C42, M4(2)⋊4C4, C82M4(2), C23.25D4, C2×C8.C4, C8.(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C428C4, C8.(C4⋊C4)

Smallest permutation representation of C8.(C4⋊C4)
On 32 points
Generators in S32
(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 12 25 22 5 16 29 18)(2 9 26 19 6 13 30 23)(3 14 27 24 7 10 31 20)(4 11 28 21 8 15 32 17)
(2 8)(3 7)(4 6)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(26 32)(27 31)(28 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E···8J8K8L8M8N
order12222444444444444488888···88888
size11222112224444888822224···48888

32 irreducible representations

dim1111111122224
type+++++++-
imageC1C2C2C2C2C2C4C4D4Q8C4○D4C4○D4C8.(C4⋊C4)
kernelC8.(C4⋊C4)C4.9C42M4(2)⋊4C4C82M4(2)C23.25D4C2×C8.C4C4×C8C8⋊C4C2×C8C2×C8C2×C4C23C1
# reps1221114422624

Matrix representation of C8.(C4⋊C4) in GL4(𝔽17) generated by

141200
121400
0035
0053
,
00145
00514
51400
14500
,
01600
1000
0004
00130
G:=sub<GL(4,GF(17))| [14,12,0,0,12,14,0,0,0,0,3,5,0,0,5,3],[0,0,5,14,0,0,14,5,14,5,0,0,5,14,0,0],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0] >;

C8.(C4⋊C4) in GAP, Magma, Sage, TeX

C_8.(C_4\rtimes C_4)
% in TeX

G:=Group("C8.(C4:C4)");
// GroupNames label

G:=SmallGroup(128,565);
// by ID

G=gap.SmallGroup(128,565);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,172,2028,1027]);
// Polycyclic

G:=Group<a,b,c|a^8=c^4=1,b^4=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^3>;
// generators/relations

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