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G = C42.102D4order 128 = 27

84th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.102D4, C43C4≀C2, D42(C4⋊C4), (C4×D4)⋊15C4, Q82(C4⋊C4), (C4×Q8)⋊15C4, C4○D4.3Q8, C4○D4.22D4, C426C413C2, C42.140(C2×C4), (C22×C4).270D4, C23.548(C2×D4), C4.181(C4⋊D4), C4⋊M4(2)⋊24C2, C4.110(C22⋊Q8), C4.35(C42⋊C2), C22.38(C4⋊D4), (C2×C42).251C22, C22.48(C22⋊Q8), (C22×C4).1328C23, C2.15(C23.7Q8), C42⋊C2.267C22, C2.39(C42⋊C22), (C2×M4(2)).153C22, (C4×C4⋊C4)⋊5C2, C4.5(C2×C4⋊C4), (C2×C4≀C2).7C2, C2.39(C2×C4≀C2), (C4×C4○D4).7C2, C4⋊C4.193(C2×C4), (C2×C4).978(C2×D4), (C2×C4).263(C2×Q8), (C2×D4).207(C2×C4), (C2×Q8).190(C2×C4), (C2×C4).864(C4○D4), (C2×C4).366(C22×C4), (C2×C4).360(C22⋊C4), (C2×C4○D4).255C22, C22.251(C2×C22⋊C4), SmallGroup(128,538)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.102D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.102D4
C1C2C2×C4 — C42.102D4
C1C2×C4C2×C42 — C42.102D4
C1C2C2C22×C4 — C42.102D4

Generators and relations for C42.102D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 292 in 156 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×2], C4 [×11], C22 [×3], C22 [×6], C8 [×2], C2×C4 [×10], C2×C4 [×22], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×7], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2.C42, C4≀C2 [×4], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, C426C4 [×2], C4×C4⋊C4, C2×C4≀C2 [×2], C4⋊M4(2), C4×C4○D4, C42.102D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C4≀C2, C42⋊C22, C42.102D4

Smallest permutation representation of C42.102D4
On 32 points
Generators in S32
(1 24 26 9)(2 10 27 17)(3 18 28 11)(4 12 29 19)(5 20 30 13)(6 14 31 21)(7 22 32 15)(8 16 25 23)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 11)(2 14 6 10)(3 13)(4 16 8 12)(5 15)(7 9)(17 27 21 31)(18 26)(19 29 23 25)(20 28)(22 30)(24 32)
(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)| (1,24,26,9)(2,10,27,17)(3,18,28,11)(4,12,29,19)(5,20,30,13)(6,14,31,21)(7,22,32,15)(8,16,25,23), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,11)(2,14,6,10)(3,13)(4,16,8,12)(5,15)(7,9)(17,27,21,31)(18,26)(19,29,23,25)(20,28)(22,30)(24,32), (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( (1,24,26,9)(2,10,27,17)(3,18,28,11)(4,12,29,19)(5,20,30,13)(6,14,31,21)(7,22,32,15)(8,16,25,23), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,11)(2,14,6,10)(3,13)(4,16,8,12)(5,15)(7,9)(17,27,21,31)(18,26)(19,29,23,25)(20,28)(22,30)(24,32), (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([(1,24,26,9),(2,10,27,17),(3,18,28,11),(4,12,29,19),(5,20,30,13),(6,14,31,21),(7,22,32,15),(8,16,25,23)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,11),(2,14,6,10),(3,13),(4,16,8,12),(5,15),(7,9),(17,27,21,31),(18,26),(19,29,23,25),(20,28),(22,30),(24,32)], [(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···4Z8A8B8C8D
order1222222244444···44···48888
size1111224411112···24···48888

38 irreducible representations

dim111111112222224
type+++++++++-
imageC1C2C2C2C2C2C4C4D4D4D4Q8C4○D4C4≀C2C42⋊C22
kernelC42.102D4C426C4C4×C4⋊C4C2×C4≀C2C4⋊M4(2)C4×C4○D4C4×D4C4×Q8C42C22×C4C4○D4C4○D4C2×C4C4C2
# reps121211442222482

Matrix representation of C42.102D4 in GL4(𝔽17) generated by

0400
4000
00130
0004
,
16000
01600
0040
0004
,
01300
4000
00160
00013
,
4000
01300
00013
00160
G:=sub<GL(4,GF(17))| [0,4,0,0,4,0,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,13,0,0,0,0,0,16,0,0,0,0,13],[4,0,0,0,0,13,0,0,0,0,0,16,0,0,13,0] >;

C42.102D4 in GAP, Magma, Sage, TeX

C_4^2._{102}D_4
% in TeX

G:=Group("C4^2.102D4");
// GroupNames label

G:=SmallGroup(128,538);
// by ID

G=gap.SmallGroup(128,538);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,718,172,124]);
// Polycyclic

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

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