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G = C42.7C8order 128 = 27

4th non-split extension by C42 of C8 acting via C8/C4=C2

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

Aliases: C42.7C8, M5(2)⋊4C4, C23.26M4(2), C4.3(C4×C8), (C2×C8).10C8, C4.24(C4⋊C8), C8.33(C4⋊C4), (C2×C8).56Q8, (C2×C8).373D4, (C2×C4).53C42, (C2×C42).42C4, (C22×C8).24C4, C2.3(C8.C8), C22.19(C4⋊C8), C4.20(C22⋊C8), C8.55(C22⋊C4), (C2×M5(2)).7C2, (C2×C4).69M4(2), C22.3(C8⋊C4), C22.23(C22⋊C8), (C22×C8).571C22, C4.26(C2.C42), C2.14(C22.7C42), (C2×C4×C8).57C2, (C2×C4).74(C2×C8), (C2×C8).188(C2×C4), (C2×C4).159(C4⋊C4), (C22×C4).467(C2×C4), (C2×C4).347(C22⋊C4), SmallGroup(128,108)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.7C8
C1C2C4C2×C4C2×C8C22×C8C2×C4×C8 — C42.7C8
C1C2C4 — C42.7C8
C1C2×C8C22×C8 — C42.7C8
C1C2C2C2C2C4C4C22×C8 — C42.7C8

Generators and relations for C42.7C8
 G = < a,b,c | a4=b4=1, c8=b2, cac-1=ab=ba, cbc-1=b-1 >

Subgroups: 104 in 74 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×6], C23, C16 [×4], C42 [×2], C42, C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4, C4×C8 [×2], C2×C16 [×2], M5(2) [×4], M5(2) [×2], C2×C42, C22×C8 [×2], C2×C4×C8, C2×M5(2) [×2], C42.7C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, C8.C8 [×2], C42.7C8

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T16A···16P
order12222244444···48···88···816···16
size11112211112···21···12···24···4

56 irreducible representations

dim1111111122222
type++++-
imageC1C2C2C4C4C4C8C8D4Q8M4(2)M4(2)C8.C8
kernelC42.7C8C2×C4×C8C2×M5(2)M5(2)C2×C42C22×C8C42C2×C8C2×C8C2×C8C2×C4C23C2
# reps11282288312216

Matrix representation of C42.7C8 in GL3(𝔽17) generated by

100
0130
001
,
100
040
0013
,
1500
001
0150
G:=sub<GL(3,GF(17))| [1,0,0,0,13,0,0,0,1],[1,0,0,0,4,0,0,0,13],[15,0,0,0,0,15,0,1,0] >;

C42.7C8 in GAP, Magma, Sage, TeX

C_4^2._7C_8
% in TeX

G:=Group("C4^2.7C8");
// GroupNames label

G:=SmallGroup(128,108);
// by ID

G=gap.SmallGroup(128,108);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,1430,352,136,2804,124]);
// Polycyclic

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

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