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G = D4×C4○D4order 128 = 27

Direct product of D4 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4×C4○D4, C23.25C24, C22.57C25, C24.494C23, C42.556C23, C4D42, C4(D4×Q8), D4226C2, D42(C4×D4), Q82(C4×D4), D412(C2×D4), (D4×Q8)⋊33C2, Q812(C2×D4), C4(D46D4), C4(D45D4), C4(Q85D4), C4(Q86D4), C4⋊Q883C22, D46D446C2, D45D442C2, Q85D436C2, Q86D432C2, (C4×Q8)⋊93C22, C2.24(D4×C23), (C4×D4)⋊105C22, C4⋊C4.469C23, C41D448C22, C4⋊D473C22, (C2×C4).599C24, (C23×C4)⋊36C22, (C2×C42)⋊52C22, C4.113(C22×D4), C22⋊Q886C22, C22≀C232C22, C22.8(C22×D4), (C2×D4).453C23, C4.4D472C22, (C22×D4)⋊63C22, C22⋊C4.13C23, (C2×Q8).431C23, (C22×Q8)⋊64C22, C22.19C2417C2, C42⋊C293C22, C2.10(C2.C25), (C22×C4).1194C23, C22.26C2429C2, C22.D442C22, (C2×C4)D42, (C2×C4)(D4×Q8), (C2×Q8)(C4×D4), (C2×C4×D4)⋊83C2, C44(C2×C4○D4), C4⋊C42(C4○D4), (C2×C4)⋊19(C2×D4), (C4×C4○D4)⋊20C2, C224(C2×C4○D4), (C2×C4)(D46D4), (C2×C4)(D45D4), C22⋊C42(C4○D4), (C2×C4)(Q86D4), (C2×C4⋊C4)⋊134C22, (C2×C4○D4)⋊74C22, (C22×C4○D4)⋊20C2, C2.29(C22×C4○D4), (C2×C22⋊C4)⋊87C22, C4⋊C4(C2×C4○D4), (C4×D4)(C2×C4○D4), (C2×D4)(C2×C4○D4), C22⋊C4(C2×C4○D4), SmallGroup(128,2200)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D4×C4○D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — D4×C4○D4
C1C22 — D4×C4○D4
C1C2×C4 — D4×C4○D4
C1C22 — D4×C4○D4

Generators and relations for D4×C4○D4
 G = < a,b,c,d,e | a4=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1300 in 824 conjugacy classes, 438 normal (20 characteristic)
C1, C2 [×3], C2 [×16], C4 [×10], C4 [×15], C22, C22 [×10], C22 [×52], C2×C4 [×3], C2×C4 [×29], C2×C4 [×61], D4 [×16], D4 [×60], Q8 [×4], Q8 [×12], C23 [×11], C23 [×30], C42, C42 [×9], C22⋊C4 [×38], C4⋊C4, C4⋊C4 [×21], C22×C4 [×35], C22×C4 [×24], C2×D4, C2×D4 [×36], C2×D4 [×36], C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C4○D4 [×8], C4○D4 [×56], C24 [×6], C2×C42 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×33], C4×Q8 [×2], C22≀C2 [×12], C4⋊D4 [×24], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4 [×6], C41D4 [×3], C4⋊Q8 [×3], C23×C4 [×6], C22×D4 [×9], C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×14], C2×C4○D4 [×16], C2×C4×D4 [×3], C4×C4○D4, C22.19C24 [×6], C22.26C24 [×3], D42 [×3], D45D4 [×6], D46D4 [×3], Q85D4 [×2], D4×Q8, Q86D4, C22×C4○D4 [×2], D4×C4○D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×4], C24 [×31], C22×D4 [×14], C2×C4○D4 [×6], C25, D4×C23, C22×C4○D4, C2.C25, D4×C4○D4

Smallest permutation representation of D4×C4○D4
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)
(2 4)(5 7)(9 11)(13 15)(17 19)(22 24)(26 28)(30 32)
(1 14 25 21)(2 15 26 22)(3 16 27 23)(4 13 28 24)(5 11 17 30)(6 12 18 31)(7 9 19 32)(8 10 20 29)
(1 21 25 14)(2 22 26 15)(3 23 27 16)(4 24 28 13)(5 11 17 30)(6 12 18 31)(7 9 19 32)(8 10 20 29)
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)(17 28)(18 25)(19 26)(20 27)(21 31)(22 32)(23 29)(24 30)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D···2M2N···2S4A4B4C4D4E···4R4S···4AD
order12222···22···244444···44···4
size11112···24···411112···24···4

50 irreducible representations

dim111111111111224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C2.C25
kernelD4×C4○D4C2×C4×D4C4×C4○D4C22.19C24C22.26C24D42D45D4D46D4Q85D4D4×Q8Q86D4C22×C4○D4C4○D4D4C2
# reps131633632112882

Matrix representation of D4×C4○D4 in GL4(𝔽5) generated by

1200
4400
0010
0001
,
1000
4400
0010
0001
,
1000
0100
0030
0003
,
1000
0100
0020
0003
,
1000
0100
0003
0020
G:=sub<GL(4,GF(5))| [1,4,0,0,2,4,0,0,0,0,1,0,0,0,0,1],[1,4,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0] >;

D4×C4○D4 in GAP, Magma, Sage, TeX

D_4\times C_4\circ D_4
% in TeX

G:=Group("D4xC4oD4");
// GroupNames label

G:=SmallGroup(128,2200);
// by ID

G=gap.SmallGroup(128,2200);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,102]);
// Polycyclic

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

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