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G = C42.401D4order 128 = 27

34th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.401D4, C42.606C23, C4.23C4≀C2, C4⋊Q8.9C4, Q8⋊C829C2, C22⋊Q8.4C4, C4.29(C8○D4), C42.59(C2×C4), (C4×Q8).4C22, C4⋊C8.252C22, (C4×C8).311C22, (C22×C4).203D4, (C4×M4(2)).16C2, C4.126(C8.C22), C23.45(C22⋊C4), (C2×C42).162C22, C42.12C4.17C2, C2.4(C23.38D4), C23.37C23.2C2, C2.8(C2×C4≀C2), C4⋊C4.52(C2×C4), (C2×Q8).46(C2×C4), (C2×C4).1448(C2×D4), (C22×C4).184(C2×C4), (C2×C4).311(C22×C4), (C2×C4).166(C22⋊C4), C22.161(C2×C22⋊C4), C2.17((C22×C8)⋊C2), SmallGroup(128,217)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.401D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.401D4
C1C2C2×C4 — C42.401D4
C1C2×C4C2×C42 — C42.401D4
C1C22C22C42 — C42.401D4

Generators and relations for C42.401D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, ad=da, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 109 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×7], C22, C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×10], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8 [×2], C4×C8, C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×M4(2), Q8⋊C8 [×4], C4×M4(2), C42.12C4, C23.37C23, C42.401D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C8○D4 [×2], C8.C22 [×2], (C22×C8)⋊C2, C23.38D4, C2×C4≀C2, C42.401D4

Smallest permutation representation of C42.401D4
On 64 points
Generators in S64
(1 23 59 15)(2 20 60 12)(3 17 61 9)(4 22 62 14)(5 19 63 11)(6 24 64 16)(7 21 57 13)(8 18 58 10)(25 44 56 37)(26 41 49 34)(27 46 50 39)(28 43 51 36)(29 48 52 33)(30 45 53 38)(31 42 54 35)(32 47 55 40)
(1 17 63 13)(2 18 64 14)(3 19 57 15)(4 20 58 16)(5 21 59 9)(6 22 60 10)(7 23 61 11)(8 24 62 12)(25 39 52 42)(26 40 53 43)(27 33 54 44)(28 34 55 45)(29 35 56 46)(30 36 49 47)(31 37 50 48)(32 38 51 41)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 33 17 54 63 44 13 27)(2 30 18 36 64 49 14 47)(3 42 19 25 57 39 15 52)(4 55 20 45 58 28 16 34)(5 37 21 50 59 48 9 31)(6 26 22 40 60 53 10 43)(7 46 23 29 61 35 11 56)(8 51 24 41 62 32 12 38)

G:=sub<Sym(64)| (1,23,59,15)(2,20,60,12)(3,17,61,9)(4,22,62,14)(5,19,63,11)(6,24,64,16)(7,21,57,13)(8,18,58,10)(25,44,56,37)(26,41,49,34)(27,46,50,39)(28,43,51,36)(29,48,52,33)(30,45,53,38)(31,42,54,35)(32,47,55,40), (1,17,63,13)(2,18,64,14)(3,19,57,15)(4,20,58,16)(5,21,59,9)(6,22,60,10)(7,23,61,11)(8,24,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33,17,54,63,44,13,27)(2,30,18,36,64,49,14,47)(3,42,19,25,57,39,15,52)(4,55,20,45,58,28,16,34)(5,37,21,50,59,48,9,31)(6,26,22,40,60,53,10,43)(7,46,23,29,61,35,11,56)(8,51,24,41,62,32,12,38)>;

G:=Group( (1,23,59,15)(2,20,60,12)(3,17,61,9)(4,22,62,14)(5,19,63,11)(6,24,64,16)(7,21,57,13)(8,18,58,10)(25,44,56,37)(26,41,49,34)(27,46,50,39)(28,43,51,36)(29,48,52,33)(30,45,53,38)(31,42,54,35)(32,47,55,40), (1,17,63,13)(2,18,64,14)(3,19,57,15)(4,20,58,16)(5,21,59,9)(6,22,60,10)(7,23,61,11)(8,24,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33,17,54,63,44,13,27)(2,30,18,36,64,49,14,47)(3,42,19,25,57,39,15,52)(4,55,20,45,58,28,16,34)(5,37,21,50,59,48,9,31)(6,26,22,40,60,53,10,43)(7,46,23,29,61,35,11,56)(8,51,24,41,62,32,12,38) );

G=PermutationGroup([(1,23,59,15),(2,20,60,12),(3,17,61,9),(4,22,62,14),(5,19,63,11),(6,24,64,16),(7,21,57,13),(8,18,58,10),(25,44,56,37),(26,41,49,34),(27,46,50,39),(28,43,51,36),(29,48,52,33),(30,45,53,38),(31,42,54,35),(32,47,55,40)], [(1,17,63,13),(2,18,64,14),(3,19,57,15),(4,20,58,16),(5,21,59,9),(6,22,60,10),(7,23,61,11),(8,24,62,12),(25,39,52,42),(26,40,53,43),(27,33,54,44),(28,34,55,45),(29,35,56,46),(30,36,49,47),(31,37,50,48),(32,38,51,41)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,33,17,54,63,44,13,27),(2,30,18,36,64,49,14,47),(3,42,19,25,57,39,15,52),(4,55,20,45,58,28,16,34),(5,37,21,50,59,48,9,31),(6,26,22,40,60,53,10,43),(7,46,23,29,61,35,11,56),(8,51,24,41,62,32,12,38)])

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4L4M4N4O4P4Q8A···8P
order1222244444···4444448···8
size1111411112···2488884···4

38 irreducible representations

dim111111122224
type+++++++-
imageC1C2C2C2C2C4C4D4D4C4≀C2C8○D4C8.C22
kernelC42.401D4Q8⋊C8C4×M4(2)C42.12C4C23.37C23C22⋊Q8C4⋊Q8C42C22×C4C4C4C4
# reps141114422882

Matrix representation of C42.401D4 in GL4(𝔽17) generated by

1000
01600
0040
0004
,
13000
01300
0040
0004
,
0900
9000
00160
0004
,
9000
0900
00013
00160
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[0,9,0,0,9,0,0,0,0,0,16,0,0,0,0,4],[9,0,0,0,0,9,0,0,0,0,0,16,0,0,13,0] >;

C42.401D4 in GAP, Magma, Sage, TeX

C_4^2._{401}D_4
% in TeX

G:=Group("C4^2.401D4");
// GroupNames label

G:=SmallGroup(128,217);
// by ID

G=gap.SmallGroup(128,217);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1059,184,1123,570,136,172]);
// Polycyclic

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

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