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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⋊C4⋊8C4, C8.4(C4⋊C4), (C2×C8).9Q8, (C2×C8).107D4, C42.21(C2×C4), C4.9C42.3C2, C23.5(C4○D4), C2.6(C42⋊8C4), C4.47(C4.4D4), C8○2M4(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

Derived series
C1 — C2×C4 — C8.(C4⋊C4)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2) — C8.(C4⋊C4)
Lower central
C1 — C2 — C2×C4 — C8.(C4⋊C4)
Upper central
C1 — C4 — C42⋊C2 — C8.(C4⋊C4)
Jennings
C1 — C2 — C2 — C22×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 [×3], C4 [×4], C4 [×4], C22 [×3], C22, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×4], C23, C42 [×2], C42 [×2], C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×6], C2×C8 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C4.Q8, C2.D8, C8.C4 [×2], C42⋊C2, C42⋊C2 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C4.9C42 [×2], M4(2)⋊4C4 [×2], C8○2M4(2), C23.25D4, C2×C8.C4, C8.(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C42⋊8C4, 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 14 25 22 5 10 29 18)(2 11 26 19 6 15 30 23)(3 16 27 24 7 12 31 20)(4 13 28 21 8 9 32 17)

(2 8)(3 7)(4 6)(9 21 13 17)(10 20 14 24)(11 19 15 23)(12 18 16 22)(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), (1,14,25,22,5,10,29,18)(2,11,26,19,6,15,30,23)(3,16,27,24,7,12,31,20)(4,13,28,21,8,9,32,17), (2,8)(3,7)(4,6)(9,21,13,17)(10,20,14,24)(11,19,15,23)(12,18,16,22)(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), (1,14,25,22,5,10,29,18)(2,11,26,19,6,15,30,23)(3,16,27,24,7,12,31,20)(4,13,28,21,8,9,32,17), (2,8)(3,7)(4,6)(9,21,13,17)(10,20,14,24)(11,19,15,23)(12,18,16,22)(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)], [(1,14,25,22,5,10,29,18),(2,11,26,19,6,15,30,23),(3,16,27,24,7,12,31,20),(4,13,28,21,8,9,32,17)], [(2,8),(3,7),(4,6),(9,21,13,17),(10,20,14,24),(11,19,15,23),(12,18,16,22),(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)⋊4C4C8○2M4(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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