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G = C8.14C42order 128 = 27

8th non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.14C42, (C4×C8)⋊19C4, C4.1(C4×Q8), C82(C4.Q8), C83(C2.D8), C4.Q814C4, C2.D814C4, C8.44(C4⋊C4), (C2×C8).60Q8, C2.5(C8○D8), C4.171(C4×D4), (C2×C8).401D4, C82(C8.C4), C8.C410C4, C22.1(C4×Q8), C4.21(C2×C42), C22.93(C4×D4), C82(C426C4), C42.315(C2×C4), C426C4.15C2, C8(C23.25D4), M4(2).19(C2×C4), C23.201(C4○D4), C82M4(2).15C2, (C22×C8).547C22, (C22×C4).1313C23, (C2×C42).1050C22, C23.25D4.21C2, C42⋊C2.265C22, C22.23(C42⋊C2), (C2×M4(2)).310C22, (C2×C4×C8).49C2, C4.76(C2×C4⋊C4), C2.11(C4×C4⋊C4), C8(C2×C8.C4), (C2×C8)(C2.D8), (C2×C8)(C4.Q8), C4⋊C4.144(C2×C4), (C2×C8)(C8.C4), (C2×C8).244(C2×C4), (C2×C4).182(C2×Q8), (C2×C8)(C426C4), (C2×C4).1507(C2×D4), (C2×C8.C4).26C2, (C2×C4).544(C4○D4), (C2×C4).522(C22×C4), (C2×C8)(C23.25D4), SmallGroup(128,504)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.14C42
C1C2C22C23C22×C4C22×C8C82M4(2) — C8.14C42
C1C2C4 — C8.14C42
C1C2×C8C22×C8 — C8.14C42
C1C2C2C22×C4 — C8.14C42

Generators and relations for C8.14C42
 G = < a,b,c | a8=b4=c4=1, bab-1=a3, ac=ca, cbc-1=a2b >

Subgroups: 164 in 116 conjugacy classes, 76 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×2], C8 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×10], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×8], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4, C4×C8 [×4], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C8.C4 [×4], C2×C42, C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C426C4 [×2], C2×C4×C8, C82M4(2) [×2], C23.25D4, C2×C8.C4, C8.14C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×C4⋊C4, C8○D8 [×2], C8.14C42

Smallest permutation representation of C8.14C42
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 11 19 26)(2 14 20 29)(3 9 21 32)(4 12 22 27)(5 15 23 30)(6 10 24 25)(7 13 17 28)(8 16 18 31)
(9 11 13 15)(10 12 14 16)(25 27 29 31)(26 28 30 32)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O···4V8A···8H8I···8T8U···8AB
order12222244444···44···48···88···88···8
size11112211112···24···41···12···24···4

56 irreducible representations

dim111111111122222
type+++++++-
imageC1C2C2C2C2C2C4C4C4C4D4Q8C4○D4C4○D4C8○D8
kernelC8.14C42C426C4C2×C4×C8C82M4(2)C23.25D4C2×C8.C4C4×C8C4.Q8C2.D8C8.C4C2×C8C2×C8C2×C4C23C2
# reps1212118448222216

Matrix representation of C8.14C42 in GL3(𝔽17) generated by

1600
020
058
,
400
0415
0013
,
1600
010
01013
G:=sub<GL(3,GF(17))| [16,0,0,0,2,5,0,0,8],[4,0,0,0,4,0,0,15,13],[16,0,0,0,1,10,0,0,13] >;

C8.14C42 in GAP, Magma, Sage, TeX

C_8._{14}C_4^2
% in TeX

G:=Group("C8.14C4^2");
// GroupNames label

G:=SmallGroup(128,504);
// by ID

G=gap.SmallGroup(128,504);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^2*b>;
// generators/relations

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