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G = C428D4order 128 = 27

2nd semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C428D4, C4⋊Q819C4, C4.16(C4×D4), C41D415C4, C424C47C2, C4.4D417C4, C4.51(C41D4), C42.157(C2×C4), (C22×C4).299D4, C23.570(C2×D4), C4⋊M4(2)⋊27C2, C4.91(C4.4D4), C22.17(C4⋊D4), (C2×C42).318C22, (C22×C4).1407C23, C2.43(C42⋊C22), (C2×M4(2)).207C22, C22.26C24.21C2, C2.15(C24.3C22), (C2×C4≀C2)⋊19C2, (C2×C4).741(C2×D4), (C2×Q8).93(C2×C4), (C2×D4).108(C2×C4), (C2×C4).598(C4○D4), (C2×C4).421(C22×C4), (C2×C4○D4).38C22, (C2×C4).139(C22⋊C4), C22.285(C2×C22⋊C4), SmallGroup(128,695)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C428D4
C1C2C4C2×C4C22×C4C2×C42C424C4 — C428D4
C1C2C2×C4 — C428D4
C1C2×C4C2×C42 — C428D4
C1C2C2C22×C4 — C428D4

Generators and relations for C428D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b, bc=cb, bd=db, dcd=c-1 >

Subgroups: 340 in 160 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×6], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C2.C42 [×2], C4≀C2 [×8], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C424C4, C2×C4≀C2 [×4], C4⋊M4(2), C22.26C24, C428D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C42⋊C22 [×2], C428D4

Smallest permutation representation of C428D4
On 32 points
Generators in S32
(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 21 19 28)(2 22 20 25)(3 23 17 26)(4 24 18 27)(5 32 13 11)(6 29 14 12)(7 30 15 9)(8 31 16 10)
(1 24 3 22)(2 28 4 26)(5 8 15 14)(6 13 16 7)(9 12 32 31)(10 30 29 11)(17 25 19 27)(18 23 20 21)
(1 6)(2 32)(3 16)(4 9)(5 25)(7 24)(8 17)(10 23)(11 20)(12 28)(13 22)(14 19)(15 27)(18 30)(21 29)(26 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4R4S4T8A8B8C8D
order122222224444444···4448888
size111122881111224···4888888

32 irreducible representations

dim111111112224
type+++++++
imageC1C2C2C2C2C4C4C4D4D4C4○D4C42⋊C22
kernelC428D4C424C4C2×C4≀C2C4⋊M4(2)C22.26C24C4.4D4C41D4C4⋊Q8C42C22×C4C2×C4C2
# reps114114226244

Matrix representation of C428D4 in GL6(𝔽17)

040000
1300000
000400
0013000
000001
0000160
,
1600000
0160000
004000
000400
000040
000004
,
0160000
100000
000100
001000
0000016
0000160
,
1140000
460000
0000130
000004
004000
0001300

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[11,4,0,0,0,0,4,6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,13,0,0,0,0,0,0,4,0,0] >;

C428D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_8D_4
% in TeX

G:=Group("C4^2:8D4");
// GroupNames label

G:=SmallGroup(128,695);
// by ID

G=gap.SmallGroup(128,695);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248,124]);
// Polycyclic

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

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