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G = C42.405D4order 128 = 27

38th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.405D4, C42.143C23, C4.26C4≀C2, C4⋊Q8.14C4, C41D4.10C4, C42.84(C2×C4), (C4×M4(2))⋊17C2, (C22×C4).223D4, C4.23(C4.D4), C8⋊C4.143C22, C23.56(C22⋊C4), (C2×C42).187C22, C42(C42.C22), C42.C2217C2, C4.4D4.112C22, C22.26C24.9C2, C2.30(C2×C4≀C2), (C2×C4○D4).3C4, (C2×D4).19(C2×C4), (C2×Q8).19(C2×C4), (C2×C4).1171(C2×D4), C2.10(C2×C4.D4), (C2×C4).137(C22×C4), (C22×C4).209(C2×C4), (C2×C4).243(C22⋊C4), C22.201(C2×C22⋊C4), (C2×C4)(C42.C22), SmallGroup(128,257)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.405D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.405D4
C1C22C2×C4 — C42.405D4
C1C2×C4C2×C42 — C42.405D4
C1C22C22C42 — C42.405D4

Generators and relations for C42.405D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 292 in 135 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×6], C4 [×5], C22, C22 [×9], C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×4], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C42.C22 [×4], C4×M4(2) [×2], C22.26C24, C42.405D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C2×C4.D4, C2×C4≀C2 [×2], C42.405D4

Smallest permutation representation of C42.405D4
On 64 points
Generators in S64
(1 45 39 30)(2 46 40 31)(3 47 33 32)(4 48 34 25)(5 41 35 26)(6 42 36 27)(7 43 37 28)(8 44 38 29)(9 55 19 61)(10 56 20 62)(11 49 21 63)(12 50 22 64)(13 51 23 57)(14 52 24 58)(15 53 17 59)(16 54 18 60)
(1 58 35 56)(2 63 36 53)(3 60 37 50)(4 57 38 55)(5 62 39 52)(6 59 40 49)(7 64 33 54)(8 61 34 51)(9 25 23 44)(10 30 24 41)(11 27 17 46)(12 32 18 43)(13 29 19 48)(14 26 20 45)(15 31 21 42)(16 28 22 47)
(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 25 58 23 35 44 56 9)(2 18 63 43 36 12 53 32)(3 46 60 11 37 27 50 17)(4 14 57 26 38 20 55 45)(5 29 62 19 39 48 52 13)(6 22 59 47 40 16 49 28)(7 42 64 15 33 31 54 21)(8 10 61 30 34 24 51 41)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4L4M4N4O8A···8P
order122222244444···44448···8
size111148811112···24884···4

38 irreducible representations

dim11111112224
type+++++++
imageC1C2C2C2C4C4C4D4D4C4≀C2C4.D4
kernelC42.405D4C42.C22C4×M4(2)C22.26C24C41D4C4⋊Q8C2×C4○D4C42C22×C4C4C4
# reps142122422162

Matrix representation of C42.405D4 in GL4(𝔽17) generated by

13000
01300
0040
0004
,
0100
1000
0040
0004
,
6700
101100
00012
001112
,
11700
71100
0005
00110
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[6,10,0,0,7,11,0,0,0,0,0,11,0,0,12,12],[11,7,0,0,7,11,0,0,0,0,0,11,0,0,5,0] >;

C42.405D4 in GAP, Magma, Sage, TeX

C_4^2._{405}D_4
% in TeX

G:=Group("C4^2.405D4");
// GroupNames label

G:=SmallGroup(128,257);
// by ID

G=gap.SmallGroup(128,257);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,1018,248,1971,102]);
// 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,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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