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G = C42.232D4order 128 = 27

214th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.232D4, C42.348C23, D8⋊C47C2, C4⋊C813C22, (C4×D4)⋊8C22, D4.Q818C2, (C4×Q8)⋊8C22, C8⋊C44C22, D4.7(C4○D4), C22⋊D8.3C2, C22⋊SD165C2, C4⋊C4.67C23, (C2×C8).41C23, C4.Q813C22, C2.D824C22, SD16⋊C48C2, D4.2D419C2, C42.6C45C2, (C2×C4).312C24, (C2×D8).59C22, (C22×C4).452D4, C23.676(C2×D4), (C2×Q8).78C23, D4⋊C422C22, Q8⋊C424C22, (C2×D4).405C23, C22.D816C2, C23.47D45C2, C4.4D455C22, C22⋊C8.25C22, C42.C232C22, C4⋊D4.167C22, C22.30(C8⋊C22), (C2×C42).839C22, (C2×SD16).13C22, C22.572(C22×D4), C22⋊Q8.172C22, C2.31(D8⋊C22), C23.36C234C2, (C22×C4).1028C23, (C22×D4).577C22, C2.113(C22.19C24), (C2×C4×D4)⋊65C2, C4.197(C2×C4○D4), C2.35(C2×C8⋊C22), (C2×C4).1220(C2×D4), (C2×C4⋊C4).940C22, SmallGroup(128,1846)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.232D4
C1C2C4C2×C4C42C4×D4C2×C4×D4 — C42.232D4
C1C2C2×C4 — C42.232D4
C1C22C2×C42 — C42.232D4
C1C2C2C2×C4 — C42.232D4

Generators and relations for C42.232D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b, dcd=a2c3 >

Subgroups: 460 in 225 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×21], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×11], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C24, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×4], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C42.6C4, SD16⋊C4 [×2], D8⋊C4 [×2], C22⋊D8, C22⋊SD16, D4.2D4 [×2], D4.Q8 [×2], C22.D8, C23.47D4, C2×C4×D4, C23.36C23, C42.232D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, D8⋊C22, C42.232D4

Smallest permutation representation of C42.232D4
On 32 points
Generators in S32
(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 15 13 11)(10 21 14 17)(12 23 16 19)(18 24 22 20)(26 32 30 28)
(1 16 31 21)(2 13 32 18)(3 10 25 23)(4 15 26 20)(5 12 27 17)(6 9 28 22)(7 14 29 19)(8 11 30 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4H4I···4N4O4P4Q8A8B8C8D
order122222222224···44···44448888
size111122444482···24···48888888

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22D8⋊C22
kernelC42.232D4C42.6C4SD16⋊C4D8⋊C4C22⋊D8C22⋊SD16D4.2D4D4.Q8C22.D8C23.47D4C2×C4×D4C23.36C23C42C22×C4D4C22C2
# reps11221122111122822

Matrix representation of C42.232D4 in GL6(𝔽17)

1600000
010000
0001600
001000
001414016
0014310
,
1300000
0130000
0013000
0001300
0051240
00121204
,
010000
1600000
0014320
001414015
000033
0000143
,
0160000
1600000
00314150
001414015
0000143
000033

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

C42.232D4 in GAP, Magma, Sage, TeX

C_4^2._{232}D_4
% in TeX

G:=Group("C4^2.232D4");
// GroupNames label

G:=SmallGroup(128,1846);
// by ID

G=gap.SmallGroup(128,1846);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,304,4037,1027,124]);
// Polycyclic

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

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