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G = C42.370D4order 128 = 27

3rd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.370D4, C4⋊C8.3C4, C4.39C4≀C2, C8⋊C4.10C4, C22⋊C8.10C4, (C2×C4).14C42, C42.40(C2×C4), (C22×C4).25Q8, C23.45(C4⋊C4), (C22×C4).643D4, C2.13(C426C4), C42.6C4.10C2, C22.7(C8.C4), C2.9(C4.C42), (C2×C42).137C22, C42.12C4.11C2, C2.10(M4(2)⋊4C4), C22.52(C2.C42), (C2×C4).22(C4⋊C4), (C2×C8⋊C4).16C2, (C22×C4).157(C2×C4), (C2×C4).375(C22⋊C4), SmallGroup(128,34)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.370D4
C1C2C22C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.370D4
C1C22C2×C4 — C42.370D4
C1C2×C4C2×C42 — C42.370D4
C1C22C22C2×C42 — C42.370D4

Generators and relations for C42.370D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=ba=ab, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=ab2c3 >

Subgroups: 120 in 72 conjugacy classes, 34 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×5], C23, C42 [×4], C2×C8 [×10], C22×C4 [×3], C4×C8, C8⋊C4 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C22×C8, C2×C8⋊C4, C42.12C4, C42.6C4, C42.370D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×2], C8.C4 [×2], C426C4, C4.C42, M4(2)⋊4C4, C42.370D4

Smallest permutation representation of C42.370D4
On 64 points
Generators in S64
(1 47 27 12)(2 44 28 9)(3 41 29 14)(4 46 30 11)(5 43 31 16)(6 48 32 13)(7 45 25 10)(8 42 26 15)(17 36 54 57)(18 33 55 62)(19 38 56 59)(20 35 49 64)(21 40 50 61)(22 37 51 58)(23 34 52 63)(24 39 53 60)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(25 31 29 27)(26 32 30 28)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)(41 47 45 43)(42 48 46 44)
(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 58 45 49 31 33 14 24)(2 17 42 63 32 50 11 38)(3 39 47 22 25 64 16 55)(4 56 44 36 26 23 13 61)(5 62 41 53 27 37 10 20)(6 21 46 59 28 54 15 34)(7 35 43 18 29 60 12 51)(8 52 48 40 30 19 9 57)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K4L8A···8P8Q8R8S8T
order12222244444···4448···88888
size11112211112···2444···48888

38 irreducible representations

dim1111111222224
type++++++-
imageC1C2C2C2C4C4C4D4D4Q8C4≀C2C8.C4M4(2)⋊4C4
kernelC42.370D4C2×C8⋊C4C42.12C4C42.6C4C8⋊C4C22⋊C8C4⋊C8C42C22×C4C22×C4C4C22C2
# reps1111444211882

Matrix representation of C42.370D4 in GL4(𝔽17) generated by

01300
4000
0040
0004
,
4000
0400
00160
00016
,
21400
141500
0040
0031
,
141400
31400
00215
00415
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[2,14,0,0,14,15,0,0,0,0,4,3,0,0,0,1],[14,3,0,0,14,14,0,0,0,0,2,4,0,0,15,15] >;

C42.370D4 in GAP, Magma, Sage, TeX

C_4^2._{370}D_4
% in TeX

G:=Group("C4^2.370D4");
// GroupNames label

G:=SmallGroup(128,34);
// by ID

G=gap.SmallGroup(128,34);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,172]);
// Polycyclic

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

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