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G = C42.42D4order 128 = 27

24th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.42D4, C23.17M4(2), C23⋊C8.9C2, (C2×C42).2C4, (C23×C4).16C4, C24.22(C2×C4), (C2×C4).31M4(2), C42.6C421C2, C22⋊C8.120C22, C2.8(C24.4C4), (C2×C42).148C22, C23.165(C22×C4), (C22×C4).428C23, C22.17(C2×M4(2)), C22.M4(2)⋊13C2, C2.6(C23.C23), C2.6(M4(2).8C22), (C2×C4⋊C4).33C4, (C2×C4).1125(C2×D4), (C2×C22⋊C4).35C4, (C22×C4).43(C2×C4), (C2×C4⋊C4).736C22, (C2×C42⋊C2).2C2, (C2×C4).312(C22⋊C4), C22.146(C2×C22⋊C4), (C2×C22⋊C4).403C22, SmallGroup(128,196)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.42D4
C1C2C22C2×C4C22×C4C2×C42C2×C42⋊C2 — C42.42D4
C1C2C23 — C42.42D4
C1C22C2×C42 — C42.42D4
C1C2C22C22×C4 — C42.42D4

Generators and relations for C42.42D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=b-1c3 >

Subgroups: 244 in 126 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×10], C22 [×3], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], C23, C23 [×2], C23 [×4], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C22×C4 [×6], C22×C4 [×6], C24, C8⋊C4 [×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], C42.6C4 [×2], C2×C42⋊C2, C42.42D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4, C23.C23, M4(2).8C22, C42.42D4

Smallest permutation representation of C42.42D4
On 32 points
Generators in S32
(1 11 31 18)(2 16 32 23)(3 13 25 20)(4 10 26 17)(5 15 27 22)(6 12 28 19)(7 9 29 24)(8 14 30 21)
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 11 13 15)(10 19 14 23)(12 21 16 17)(18 20 22 24)(25 27 29 31)
(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 28 3 8 5 32 7 4)(2 29 26 31 6 25 30 27)(9 14 11 23 13 10 15 19)(12 24 21 18 16 20 17 22)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4P8A···8H
order122222224···44···48···8
size111122442···24···48···8

32 irreducible representations

dim11111111122244
type++++++
imageC1C2C2C2C2C4C4C4C4D4M4(2)M4(2)C23.C23M4(2).8C22
kernelC42.42D4C23⋊C8C22.M4(2)C42.6C4C2×C42⋊C2C2×C42C2×C22⋊C4C2×C4⋊C4C23×C4C42C2×C4C23C2C2
# reps12221222244422

Matrix representation of C42.42D4 in GL6(𝔽17)

0160000
1600000
0013000
0001300
0000130
0000013
,
400000
040000
000100
001000
007101
00161010
,
2120000
5150000
00121620
0043015
00158516
00411414
,
1220000
1550000
00121620
00131402
0011551
0010243

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,7,16,0,0,1,0,1,10,0,0,0,0,0,1,0,0,0,0,1,0],[2,5,0,0,0,0,12,15,0,0,0,0,0,0,12,4,15,4,0,0,16,3,8,11,0,0,2,0,5,4,0,0,0,15,16,14],[12,15,0,0,0,0,2,5,0,0,0,0,0,0,12,13,11,10,0,0,16,14,5,2,0,0,2,0,5,4,0,0,0,2,1,3] >;

C42.42D4 in GAP, Magma, Sage, TeX

C_4^2._{42}D_4
% in TeX

G:=Group("C4^2.42D4");
// GroupNames label

G:=SmallGroup(128,196);
// by ID

G=gap.SmallGroup(128,196);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,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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