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G = C42.404D4order 128 = 27

37th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.404D4, C4⋊Q87C4, (C4×Q8)⋊2C4, C4.25C4≀C2, (C2×C4).55Q16, C42.71(C2×C4), (C2×C4).99SD16, C22.8(C2×Q16), (C22×C4).733D4, C23.493(C2×D4), C22.9(C2×SD16), C4.33(Q8⋊C4), C43(C23.31D4), C22⋊C8.164C22, (C22×C4).625C23, (C2×C42).174C22, C23.31D4.7C2, C22⋊Q8.136C22, C42.12C4.20C2, C23.37C23.5C2, C2.C42.502C22, C2.14(C23.C23), (C4×C4⋊C4).4C2, C4⋊C4.4(C2×C4), C2.20(C2×C4≀C2), (C2×Q8).5(C2×C4), C2.8(C2×Q8⋊C4), (C2×C4).1149(C2×D4), (C2×C4).88(C22⋊C4), (C2×C4).115(C22×C4), C22.179(C2×C22⋊C4), SmallGroup(128,235)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.404D4
C1C2C22C23C22×C4C2×C42C23.37C23 — C42.404D4
C1C22C2×C4 — C42.404D4
C1C2×C4C2×C42 — C42.404D4
C1C2C22C22×C4 — C42.404D4

Generators and relations for C42.404D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 228 in 120 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×11], C22 [×3], C22 [×2], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×15], Q8 [×4], C23, C42 [×4], C42 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C2.C42 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C23.31D4 [×4], C4×C4⋊C4, C42.12C4, C23.37C23, C42.404D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C2×SD16, C2×Q16, C23.C23, C2×Q8⋊C4, C2×C4≀C2, C42.404D4

Smallest permutation representation of C42.404D4
On 32 points
Generators in S32
(1 12 25 21)(2 13 26 22)(3 14 27 23)(4 15 28 24)(5 16 29 17)(6 9 30 18)(7 10 31 19)(8 11 32 20)
(1 27 5 31)(2 28 6 32)(3 29 7 25)(4 30 8 26)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(1 17 25 16)(2 11)(3 14 27 23)(4 18)(5 21 29 12)(6 15)(7 10 31 19)(8 22)(9 28)(13 32)(20 26)(24 30)
(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)| (1,12,25,21)(2,13,26,22)(3,14,27,23)(4,15,28,24)(5,16,29,17)(6,9,30,18)(7,10,31,19)(8,11,32,20), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,17,25,16)(2,11)(3,14,27,23)(4,18)(5,21,29,12)(6,15)(7,10,31,19)(8,22)(9,28)(13,32)(20,26)(24,30), (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( (1,12,25,21)(2,13,26,22)(3,14,27,23)(4,15,28,24)(5,16,29,17)(6,9,30,18)(7,10,31,19)(8,11,32,20), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,17,25,16)(2,11)(3,14,27,23)(4,18)(5,21,29,12)(6,15)(7,10,31,19)(8,22)(9,28)(13,32)(20,26)(24,30), (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([(1,12,25,21),(2,13,26,22),(3,14,27,23),(4,15,28,24),(5,16,29,17),(6,9,30,18),(7,10,31,19),(8,11,32,20)], [(1,27,5,31),(2,28,6,32),(3,29,7,25),(4,30,8,26),(9,20,13,24),(10,21,14,17),(11,22,15,18),(12,23,16,19)], [(1,17,25,16),(2,11),(3,14,27,23),(4,18),(5,21,29,12),(6,15),(7,10,31,19),(8,22),(9,28),(13,32),(20,26),(24,30)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K···4T4U4V4W4X8A···8H
order12222244444···44···444448···8
size11112211112···24···488884···4

38 irreducible representations

dim1111111222224
type+++++++-
imageC1C2C2C2C2C4C4D4D4SD16Q16C4≀C2C23.C23
kernelC42.404D4C23.31D4C4×C4⋊C4C42.12C4C23.37C23C4×Q8C4⋊Q8C42C22×C4C2×C4C2×C4C4C2
# reps1411144224482

Matrix representation of C42.404D4 in GL4(𝔽17) generated by

1000
0100
0040
0004
,
11500
11600
0040
0004
,
16000
16100
0040
00016
,
01000
121000
00016
0040
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,1,0,0,15,16,0,0,0,0,4,0,0,0,0,4],[16,16,0,0,0,1,0,0,0,0,4,0,0,0,0,16],[0,12,0,0,10,10,0,0,0,0,0,4,0,0,16,0] >;

C42.404D4 in GAP, Magma, Sage, TeX

C_4^2._{404}D_4
% in TeX

G:=Group("C4^2.404D4");
// GroupNames label

G:=SmallGroup(128,235);
// by ID

G=gap.SmallGroup(128,235);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,520,1123,1018,248,1971]);
// Polycyclic

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

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