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G = C8○D4⋊C4order 128 = 27

1st semidirect product of C8○D4 and C4 acting via C4/C2=C2

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

Aliases: C8○D41C4, D4.2(C4⋊C4), C4○D4.23D4, (C2×C8).104D4, Q8.2(C4⋊C4), (C2×D4).26Q8, (C2×Q8).19Q8, C8.47(C22⋊C4), C23.4(C4○D4), C4.22(C22⋊Q8), C4.184(C4⋊D4), M4(2).30(C2×C4), M4(2)⋊4C412C2, C23.25D416C2, C22.40(C4⋊D4), (C22×C8).216C22, C42⋊C2.3C22, C42⋊C22.2C2, (C22×C4).664C23, C22.24(C22⋊Q8), C2.23(C23.7Q8), C22.14(C42⋊C2), (C2×M4(2)).157C22, C4.6(C2×C4⋊C4), (C2×C4).4(C2×Q8), (C2×C8).60(C2×C4), (C2×C8○D4).3C2, (C2×C8.C4)⋊4C2, C4○D4.27(C2×C4), (C2×C4).231(C2×D4), C4.93(C2×C22⋊C4), (C2×C4).739(C4○D4), (C2×C4).178(C22×C4), (C2×C4○D4).258C22, SmallGroup(128,546)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8○D4⋊C4
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — C8○D4⋊C4
C1C2C2×C4 — C8○D4⋊C4
C1C4C22×C4 — C8○D4⋊C4
C1C2C2C22×C4 — C8○D4⋊C4

Generators and relations for C8○D4⋊C4
 G = < a,b,c,d | a8=c2=d4=1, b2=a4, ab=ba, ac=ca, dad-1=a-1, cbc=a4b, dbd-1=a6c, dcd-1=a2b >

Subgroups: 228 in 128 conjugacy classes, 58 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], C2×C8 [×7], M4(2) [×2], M4(2) [×9], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C4.Q8, C2.D8, C8.C4 [×2], C42⋊C2 [×2], C22×C8, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, M4(2)⋊4C4 [×2], C42⋊C22 [×2], C23.25D4, C2×C8.C4, C2×C8○D4, C8○D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C8○D4⋊C4

Smallest permutation representation of C8○D4⋊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 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E···8J8K8L8M8N
order12222224444444444488888···88888
size11222441122244888822224···48888

32 irreducible representations

dim11111112222224
type+++++++--+
imageC1C2C2C2C2C2C4D4Q8Q8D4C4○D4C4○D4C8○D4⋊C4
kernelC8○D4⋊C4M4(2)⋊4C4C42⋊C22C23.25D4C2×C8.C4C2×C8○D4C8○D4C2×C8C2×D4C2×Q8C4○D4C2×C4C23C1
# reps12211184112224

Matrix representation of C8○D4⋊C4 in GL4(𝔽17) generated by

111100
3000
001111
0030
,
00130
00013
13000
01300
,
0012
001616
161500
1100
,
01000
5000
0006
0030
G:=sub<GL(4,GF(17))| [11,3,0,0,11,0,0,0,0,0,11,3,0,0,11,0],[0,0,13,0,0,0,0,13,13,0,0,0,0,13,0,0],[0,0,16,1,0,0,15,1,1,16,0,0,2,16,0,0],[0,5,0,0,10,0,0,0,0,0,0,3,0,0,6,0] >;

C8○D4⋊C4 in GAP, Magma, Sage, TeX

C_8\circ D_4\rtimes C_4
% in TeX

G:=Group("C8oD4:C4");
// GroupNames label

G:=SmallGroup(128,546);
// by ID

G=gap.SmallGroup(128,546);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,718,172,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^2=d^4=1,b^2=a^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^4*b,d*b*d^-1=a^6*c,d*c*d^-1=a^2*b>;
// generators/relations

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