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G = C42.428D4order 128 = 27

61st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.428D4, D4⋊C42C4, Q8⋊C42C4, C4.126(C4×D4), (C2×C8).226D4, C2.17(C8○D8), C426C428C2, C22.171(C4×D4), C4.195(C4⋊D4), C4.C4220C2, C4.86(C4.4D4), C4.11(C42⋊C2), C23.207(C4○D4), (C22×C8).485C22, C23.24D4.1C2, (C22×C4).1388C23, C42⋊C2.38C22, C42.6C2222C2, (C2×C42).1066C22, C22.1(C422C2), (C2×M4(2)).200C22, C2.22(C24.C22), C22.30(C22.D4), (C2×C4×C8)⋊11C2, C4⋊C4.85(C2×C4), (C2×C4≀C2).12C2, (C2×C8).161(C2×C4), (C2×Q8).86(C2×C4), (C2×D4).101(C2×C4), (C2×C4).1345(C2×D4), (C2×C4).583(C4○D4), (C2×C4).406(C22×C4), (C2×C4○D4).35C22, (C22×C8)⋊C2.16C2, SmallGroup(128,669)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.428D4
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C42.428D4
C1C2C2×C4 — C42.428D4
C1C2×C4C22×C8 — C42.428D4
C1C2C2C22×C4 — C42.428D4

Generators and relations for C42.428D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2b2c3 >

Subgroups: 212 in 115 conjugacy classes, 48 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×4], C4 [×7], C22 [×3], C22 [×5], C8 [×6], C2×C4 [×6], C2×C4 [×11], D4 [×4], Q8 [×2], C23, C23, C42 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×6], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C4⋊C8 [×2], C2×C42, C42⋊C2, C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C426C4, C4.C42, C2×C4×C8, (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C42.6C22, C42.428D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22, C8○D8 [×2], C42.428D4

Smallest permutation representation of C42.428D4
On 32 points
Generators in S32
(1 7 5 3)(2 27)(4 29)(6 31)(8 25)(9 18)(10 16 14 12)(11 20)(13 22)(15 24)(17 23 21 19)(26 32 30 28)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(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 28 15 5 18 32 11)(2 21 29 14 6 17 25 10)(3 24 30 9 7 20 26 13)(4 23 31 16 8 19 27 12)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4N4O4P4Q8A···8P8Q8R8S8T
order122222244444···44448···88888
size111122811112···28882···28888

44 irreducible representations

dim111111111122222
type++++++++++
imageC1C2C2C2C2C2C2C2C4C4D4D4C4○D4C4○D4C8○D8
kernelC42.428D4C426C4C4.C42C2×C4×C8(C22×C8)⋊C2C23.24D4C2×C4≀C2C42.6C22D4⋊C4Q8⋊C4C42C2×C8C2×C4C23C2
# reps1111111144226216

Matrix representation of C42.428D4 in GL4(𝔽17) generated by

41500
01600
0010
00016
,
4000
0400
00160
00016
,
12300
13500
0004
00130
,
71200
91000
0001
00160
G:=sub<GL(4,GF(17))| [4,0,0,0,15,16,0,0,0,0,1,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[12,13,0,0,3,5,0,0,0,0,0,13,0,0,4,0],[7,9,0,0,12,10,0,0,0,0,0,16,0,0,1,0] >;

C42.428D4 in GAP, Magma, Sage, TeX

C_4^2._{428}D_4
% in TeX

G:=Group("C4^2.428D4");
// GroupNames label

G:=SmallGroup(128,669);
// by ID

G=gap.SmallGroup(128,669);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,2804,172,124]);
// Polycyclic

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

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