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G = C42.444D4order 128 = 27

77th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.444D4, C42.323C23, C4○D49D4, D42(C2×D4), Q82(C2×D4), C4⋊C83C22, C4⋊D818C2, C45(C8⋊C22), C4⋊SD162C2, (C4×D4)⋊83C22, (C2×D8)⋊16C22, (C2×C8).13C23, (C4×Q8)⋊79C22, C4.69(C22×D4), C4⋊C4.379C23, C4⋊M4(2)⋊4C2, (C2×C4).242C24, (C2×SD16)⋊7C22, (C2×D4).51C23, C23.654(C2×D4), (C22×C4).422D4, C4.104(C4⋊D4), D4⋊C416C22, C23.37D46C2, (C2×Q8).359C23, C41D4.138C22, C22.77(C4⋊D4), (C22×C4).972C23, (C2×C42).811C22, C22.502(C22×D4), (C22×D4).338C22, (C2×M4(2)).49C22, C42⋊C2.311C22, (C4×C4○D4)⋊7C2, (C2×C41D4)⋊16C2, (C2×C8⋊C22)⋊15C2, C4.152(C2×C4○D4), C2.60(C2×C4⋊D4), C2.16(C2×C8⋊C22), (C2×C4).1421(C2×D4), (C2×C4).273(C4○D4), (C2×C4○D4).298C22, SmallGroup(128,1770)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.444D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.444D4
C1C2C2×C4 — C42.444D4
C1C22C2×C42 — C42.444D4
C1C2C2C2×C4 — C42.444D4

Generators and relations for C42.444D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c3 >

Subgroups: 676 in 292 conjugacy classes, 104 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×26], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×2], D4 [×33], Q8 [×2], Q8, C23, C23 [×17], C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×4], C2×D4 [×27], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24 [×2], D4⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C41D4 [×4], C41D4 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×4], C8⋊C22 [×8], C22×D4 [×2], C22×D4 [×2], C2×C4○D4, C23.37D4 [×2], C4⋊M4(2), C4⋊D8 [×4], C4⋊SD16 [×4], C4×C4○D4, C2×C41D4, C2×C8⋊C22 [×2], C42.444D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22 [×2], C42.444D4

Smallest permutation representation of C42.444D4
On 32 points
Generators in S32
(1 18 30 12)(2 13 31 19)(3 20 32 14)(4 15 25 21)(5 22 26 16)(6 9 27 23)(7 24 28 10)(8 11 29 17)
(1 32 5 28)(2 29 6 25)(3 26 7 30)(4 31 8 27)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4H4I···4P8A8B8C8D
order1222222222224···44···48888
size1111224488882···24···48888

32 irreducible representations

dim1111111122224
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22
kernelC42.444D4C23.37D4C4⋊M4(2)C4⋊D8C4⋊SD16C4×C4○D4C2×C41D4C2×C8⋊C22C42C22×C4C4○D4C2×C4C4
# reps1214411222444

Matrix representation of C42.444D4 in GL6(𝔽17)

13150000
040000
0016000
0001600
0000160
0000016
,
100000
010000
0001600
001000
0000016
000010
,
420000
1130000
0000016
0000160
0016000
000100
,
13150000
1640000
0016000
000100
0000016
0000160

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

C42.444D4 in GAP, Magma, Sage, TeX

C_4^2._{444}D_4
% in TeX

G:=Group("C4^2.444D4");
// GroupNames label

G:=SmallGroup(128,1770);
// by ID

G=gap.SmallGroup(128,1770);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2019,248,4037,1027,124]);
// Polycyclic

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

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