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G = C8⋊C22⋊C4order 128 = 27

2nd semidirect product of C8⋊C22 and C4 acting via C4/C2=C2

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

Aliases: C8⋊C222C4, C4○D4.6D4, C4.65(C4×D4), C4⋊C4.315D4, C426C43C2, D42(C22⋊C4), (C2×D4).276D4, M4(2)⋊5(C2×C4), Q82(C22⋊C4), (C2×Q8).217D4, C22.37(C4×D4), (C22×C4).24D4, C2.4(D44D4), C23.559(C2×D4), C4.131(C4⋊D4), C22.C424C2, C22.98C22≀C2, C2.4(D4.8D4), C23.37D421C2, C22.44(C4⋊D4), (C22×C4).681C23, C24.3C223C2, (C2×C42).282C22, (C22×D4).18C22, (C2×M4(2)).8C22, C42⋊C2.18C22, C23.33C231C2, C4.68(C22.D4), C2.22(C23.23D4), (C2×D4)⋊7(C2×C4), (C2×C4≀C2)⋊14C2, C4○D4.9(C2×C4), (C2×C8⋊C22).1C2, C4.17(C2×C22⋊C4), (C2×C4).1002(C2×D4), (C2×C4⋊C4).59C22, (C2×C4).11(C22×C4), (C2×C4).320(C4○D4), (C2×C4○D4).18C22, SmallGroup(128,615)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8⋊C22⋊C4
C1C2C4C2×C4C22×C4C2×C4○D4C23.33C23 — C8⋊C22⋊C4
C1C2C2×C4 — C8⋊C22⋊C4
C1C22C22×C4 — C8⋊C22⋊C4
C1C2C2C22×C4 — C8⋊C22⋊C4

Generators and relations for C8⋊C22⋊C4
 G = < a,b,c,d | a8=b2=c2=d4=1, bab=a3, cac=a5, dad-1=a3c, bc=cb, dbd-1=a2bc, dcd-1=a4c >

Subgroups: 428 in 185 conjugacy classes, 56 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×16], C8 [×3], C2×C4 [×6], C2×C4 [×17], D4 [×2], D4 [×11], Q8 [×2], Q8, C23, C23 [×9], C42 [×5], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×3], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, D4⋊C4 [×2], C4≀C2 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C2×D8, C2×SD16, C8⋊C22 [×4], C8⋊C22 [×2], C22×D4, C2×C4○D4, C426C4, C22.C42, C24.3C22, C23.37D4, C2×C4≀C2, C23.33C23, C2×C8⋊C22, C8⋊C22⋊C4
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], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D44D4, D4.8D4, C8⋊C22⋊C4

Smallest permutation representation of C8⋊C22⋊C4
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 7)(3 5)(4 8)(9 30)(10 25)(11 28)(12 31)(13 26)(14 29)(15 32)(16 27)(17 23)(19 21)(20 24)
(1 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)
(1 32 21 15)(2 14 22 31)(3 26 23 9)(4 16 24 25)(5 28 17 11)(6 10 18 27)(7 30 19 13)(8 12 20 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4R8A8B8C8D
order122222222244444···48888
size111122448822224···48888

32 irreducible representations

dim11111111122222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4D4D4D4C4○D4D44D4D4.8D4
kernelC8⋊C22⋊C4C426C4C22.C42C24.3C22C23.37D4C2×C4≀C2C23.33C23C2×C8⋊C22C8⋊C22C4⋊C4C22×C4C2×D4C2×Q8C4○D4C2×C4C2C2
# reps11111111822112422

Matrix representation of C8⋊C22⋊C4 in GL6(𝔽17)

16150000
110000
0000160
000001
000100
001000
,
1600000
110000
000100
001000
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
480000
0130000
000040
0000013
004000
0001300

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

C8⋊C22⋊C4 in GAP, Magma, Sage, TeX

C_8\rtimes C_2^2\rtimes C_4
% in TeX

G:=Group("C8:C2^2:C4");
// GroupNames label

G:=SmallGroup(128,615);
// by ID

G=gap.SmallGroup(128,615);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,248,2804,718,172,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^2=d^4=1,b*a*b=a^3,c*a*c=a^5,d*a*d^-1=a^3*c,b*c=c*b,d*b*d^-1=a^2*b*c,d*c*d^-1=a^4*c>;
// generators/relations

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