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G = C42.371D4order 128 = 27

4th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.371D4, C4(C23⋊C8), (C22×C4)⋊4C8, C23⋊C8.11C2, C24.19(C2×C4), C23.16(C2×C8), (C23×C4).14C4, (C2×C42).14C4, C4.22(C22⋊C8), (C2×C4).72M4(2), C22.8(C22×C8), C42.12C41C2, C22⋊C8.153C22, C23.159(C22×C4), (C2×C42).145C22, (C22×C4).422C23, C22.11(C2×M4(2)), C4(C22.M4(2)), C22.M4(2)⋊18C2, C2.1(C23.C23), C2.1(M4(2).8C22), (C2×C4⋊C4).29C4, (C2×C4).37(C2×C8), C2.6(C2×C22⋊C8), (C2×C4).1119(C2×D4), (C2×C22⋊C4).16C4, (C2×C4⋊C4).732C22, (C22×C4).470(C2×C4), (C2×C42⋊C2).1C2, C22.90(C2×C22⋊C4), (C2×C4).388(C22⋊C4), (C2×C22⋊C4).399C22, SmallGroup(128,190)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.371D4
C1C2C22C2×C4C22×C4C2×C42C2×C42⋊C2 — C42.371D4
C1C2C22 — C42.371D4
C1C2×C4C2×C42 — C42.371D4
C1C2C22C22×C4 — C42.371D4

Generators and relations for C42.371D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=b-1c3 >

Subgroups: 244 in 132 conjugacy classes, 60 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×10], C2×C4 [×16], C23, C23 [×2], C23 [×4], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C22×C4 [×6], C22×C4 [×4], C22×C4 [×4], C24, C4×C8 [×2], C22⋊C8 [×4], C4⋊C8 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C23×C4, C23⋊C8 [×2], C22.M4(2) [×2], C42.12C4 [×2], C2×C42⋊C2, C42.371D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C23.C23, M4(2).8C22, C42.371D4

Smallest permutation representation of C42.371D4
On 32 points
Generators in S32
(1 15 31 23)(2 16 32 24)(3 9 25 17)(4 10 26 18)(5 11 27 19)(6 12 28 20)(7 13 29 21)(8 14 30 22)
(1 7 5 3)(2 30 6 26)(4 32 8 28)(9 15 13 11)(10 24 14 20)(12 18 16 22)(17 23 21 19)(25 31 29 27)
(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)
(1 32 7 8 5 28 3 4)(2 29 30 27 6 25 26 31)(9 10 15 24 13 14 11 20)(12 17 18 23 16 21 22 19)

G:=sub<Sym(32)| (1,15,31,23)(2,16,32,24)(3,9,25,17)(4,10,26,18)(5,11,27,19)(6,12,28,20)(7,13,29,21)(8,14,30,22), (1,7,5,3)(2,30,6,26)(4,32,8,28)(9,15,13,11)(10,24,14,20)(12,18,16,22)(17,23,21,19)(25,31,29,27), (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), (1,32,7,8,5,28,3,4)(2,29,30,27,6,25,26,31)(9,10,15,24,13,14,11,20)(12,17,18,23,16,21,22,19)>;

G:=Group( (1,15,31,23)(2,16,32,24)(3,9,25,17)(4,10,26,18)(5,11,27,19)(6,12,28,20)(7,13,29,21)(8,14,30,22), (1,7,5,3)(2,30,6,26)(4,32,8,28)(9,15,13,11)(10,24,14,20)(12,18,16,22)(17,23,21,19)(25,31,29,27), (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), (1,32,7,8,5,28,3,4)(2,29,30,27,6,25,26,31)(9,10,15,24,13,14,11,20)(12,17,18,23,16,21,22,19) );

G=PermutationGroup([(1,15,31,23),(2,16,32,24),(3,9,25,17),(4,10,26,18),(5,11,27,19),(6,12,28,20),(7,13,29,21),(8,14,30,22)], [(1,7,5,3),(2,30,6,26),(4,32,8,28),(9,15,13,11),(10,24,14,20),(12,18,16,22),(17,23,21,19),(25,31,29,27)], [(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)], [(1,32,7,8,5,28,3,4),(2,29,30,27,6,25,26,31),(9,10,15,24,13,14,11,20),(12,17,18,23,16,21,22,19)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim11111111112244
type++++++
imageC1C2C2C2C2C4C4C4C4C8D4M4(2)C23.C23M4(2).8C22
kernelC42.371D4C23⋊C8C22.M4(2)C42.12C4C2×C42⋊C2C2×C42C2×C22⋊C4C2×C4⋊C4C23×C4C22×C4C42C2×C4C2C2
# reps122212222164422

Matrix representation of C42.371D4 in GL6(𝔽17)

100000
010000
004000
000400
000040
000004
,
1300000
0130000
000100
0016000
0011001
0006160
,
1420000
630000
0071120
00610015
00281011
0081567
,
3150000
15140000
00106150
00610015
0000711
0010117

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,11,0,0,0,1,0,0,6,0,0,0,0,0,16,0,0,0,0,1,0],[14,6,0,0,0,0,2,3,0,0,0,0,0,0,7,6,2,8,0,0,11,10,8,15,0,0,2,0,10,6,0,0,0,15,11,7],[3,15,0,0,0,0,15,14,0,0,0,0,0,0,10,6,0,1,0,0,6,10,0,0,0,0,15,0,7,11,0,0,0,15,11,7] >;

C42.371D4 in GAP, Magma, Sage, TeX

C_4^2._{371}D_4
% in TeX

G:=Group("C4^2.371D4");
// GroupNames label

G:=SmallGroup(128,190);
// by ID

G=gap.SmallGroup(128,190);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,851,172]);
// Polycyclic

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

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