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G = C8.23C42order 128 = 27

4th central extension by C8 of C42

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8.23C42, M5(2)⋊12C4, C23.11M4(2), M5(2).18C22, (C2×C16)⋊8C4, (C4×C8).9C4, C16.3(C2×C4), C8(C16⋊C4), C16⋊C46C2, C8.62(C22×C4), C4.44(C2×C42), (C2×C4).75C42, C42.23(C2×C4), C4.10(C8⋊C4), (C2×C8).383C23, C4.48(C2×M4(2)), (C2×C4).23M4(2), C42⋊C2.20C4, C22.9(C8⋊C4), (C2×M4(2)).27C4, (C2×M5(2)).20C2, C8⋊C4.149C22, C82M4(2).19C2, (C22×C8).413C22, C22.22(C2×M4(2)), C2.12(C2×C8⋊C4), (C2×C8).248(C2×C4), (C22×C4).285(C2×C4), (C2×C4).557(C22×C4), SmallGroup(128,842)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.23C42
C1C2C4C2×C4C2×C8C22×C8C82M4(2) — C8.23C42
C1C4 — C8.23C42
C1C8 — C8.23C42
C1C2C2C2C2C4C4C2×C8 — C8.23C42

Generators and relations for C8.23C42
 G = < a,b,c | a8=c4=1, b4=a2, ab=ba, ac=ca, cbc-1=a2b >

Subgroups: 100 in 80 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22, C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], C23, C16 [×8], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×4], M5(2) [×8], C42⋊C2, C22×C8, C2×M4(2), C16⋊C4 [×4], C82M4(2), C2×M5(2) [×2], C8.23C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C2×C8⋊C4, C8.23C42

Smallest permutation representation of C8.23C42
On 32 points
Generators in S32
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)
(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 22)(2 19 10 27)(3 32)(4 29 12 21)(5 26)(6 23 14 31)(7 20)(8 17 16 25)(9 30)(11 24)(13 18)(15 28)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E···8J8K8L8M8N16A···16P
order1222244444444488888···8888816···16
size1122211222444411112···244444···4

44 irreducible representations

dim111111111224
type++++
imageC1C2C2C2C4C4C4C4C4M4(2)M4(2)C8.23C42
kernelC8.23C42C16⋊C4C82M4(2)C2×M5(2)C4×C8C2×C16M5(2)C42⋊C2C2×M4(2)C2×C4C23C1
# reps141248822624

Matrix representation of C8.23C42 in GL4(𝔽17) generated by

2000
0200
0020
0002
,
0010
0001
2000
01500
,
0100
1000
0004
0040
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,0,2,0,0,0,0,15,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,4,0,0,4,0] >;

C8.23C42 in GAP, Magma, Sage, TeX

C_8._{23}C_4^2
% in TeX

G:=Group("C8.23C4^2");
// GroupNames label

G:=SmallGroup(128,842);
// by ID

G=gap.SmallGroup(128,842);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,477,120,723,1018,136,2804,124]);
// Polycyclic

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

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