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G = C42.403D4order 128 = 27

36th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.403D4, C4⋊Q86C4, (C4×D4)⋊2C4, C4.24C4≀C2, C41D46C4, (C2×C4).128D8, C42.70(C2×C4), C22.10(C2×D8), (C2×C4).115SD16, (C22×C4).732D4, C23.492(C2×D4), C4.54(D4⋊C4), C43(C22.SD16), C22.SD1624C2, C22.28(C2×SD16), C42.12C416C2, C4⋊D4.131C22, C22⋊C8.163C22, (C2×C42).173C22, (C22×C4).624C23, C22.26C24.5C2, C2.C42.501C22, C2.13(C23.C23), (C4×C4⋊C4)⋊1C2, C4⋊C4.3(C2×C4), C2.19(C2×C4≀C2), (C2×D4).5(C2×C4), C2.8(C2×D4⋊C4), (C2×C4).1148(C2×D4), (C2×C4).87(C22⋊C4), (C2×C4).114(C22×C4), C22.178(C2×C22⋊C4), SmallGroup(128,234)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.403D4
C1C2C22C23C22×C4C2×C42C22.26C24 — C42.403D4
C1C22C2×C4 — C42.403D4
C1C2×C4C2×C42 — C42.403D4
C1C2C22C22×C4 — C42.403D4

Generators and relations for C42.403D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 324 in 142 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×9], C22 [×3], C22 [×8], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×17], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C2.C42 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C22.SD16 [×4], C4×C4⋊C4, C42.12C4, C22.26C24, C42.403D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C2×D8, C2×SD16, C23.C23, C2×D4⋊C4, C2×C4≀C2, C42.403D4

Smallest permutation representation of C42.403D4
On 32 points
Generators in S32
(1 31 14 18)(2 32 15 19)(3 25 16 20)(4 26 9 21)(5 27 10 22)(6 28 11 23)(7 29 12 24)(8 30 13 17)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(2 13 15 8)(3 12)(4 6 9 11)(7 16)(17 19 30 32)(20 29)(21 23 26 28)(24 25)
(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)| (1,31,14,18)(2,32,15,19)(3,25,16,20)(4,26,9,21)(5,27,10,22)(6,28,11,23)(7,29,12,24)(8,30,13,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,13,15,8)(3,12)(4,6,9,11)(7,16)(17,19,30,32)(20,29)(21,23,26,28)(24,25), (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( (1,31,14,18)(2,32,15,19)(3,25,16,20)(4,26,9,21)(5,27,10,22)(6,28,11,23)(7,29,12,24)(8,30,13,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,13,15,8)(3,12)(4,6,9,11)(7,16)(17,19,30,32)(20,29)(21,23,26,28)(24,25), (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([(1,31,14,18),(2,32,15,19),(3,25,16,20),(4,26,9,21),(5,27,10,22),(6,28,11,23),(7,29,12,24),(8,30,13,17)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(2,13,15,8),(3,12),(4,6,9,11),(7,16),(17,19,30,32),(20,29),(21,23,26,28),(24,25)], [(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 2A2B2C2D2E2F2G4A4B4C4D4E···4J4K···4T4U4V8A···8H
order1222222244444···44···4448···8
size1111228811112···24···4884···4

38 irreducible representations

dim11111111222224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16C4≀C2C23.C23
kernelC42.403D4C22.SD16C4×C4⋊C4C42.12C4C22.26C24C4×D4C41D4C4⋊Q8C42C22×C4C2×C4C2×C4C4C2
# reps14111422224482

Matrix representation of C42.403D4 in GL4(𝔽17) generated by

4000
0400
0010
0001
,
13000
01300
00115
00116
,
1000
0400
0010
00116
,
0400
1000
00011
00311
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,4,0,0,0,0,1,1,0,0,0,16],[0,1,0,0,4,0,0,0,0,0,0,3,0,0,11,11] >;

C42.403D4 in GAP, Magma, Sage, TeX

C_4^2._{403}D_4
% in TeX

G:=Group("C4^2.403D4");
// GroupNames label

G:=SmallGroup(128,234);
// by ID

G=gap.SmallGroup(128,234);
# by ID

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

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,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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