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## G = D4.(C2×D4)  order 128 = 27

### 8th non-split extension by D4 of C2×D4 acting via C2×D4/C23=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.(C2×D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — D4.(C2×D4)
 Lower central C1 — C2 — C2×C4 — D4.(C2×D4)
 Upper central C1 — C22 — C2×C4○D4 — D4.(C2×D4)
 Jennings C1 — C2 — C2 — C2×C4 — D4.(C2×D4)

Generators and relations for D4.(C2×D4)
G = < a,b,c,d,e | a4=b2=1, c2=d4=e2=a2, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=ebe-1=a-1b, cd=dc, ece-1=a2c, ede-1=d3 >

Subgroups: 796 in 376 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×32], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×23], D4 [×6], D4 [×33], Q8 [×2], Q8 [×9], C23, C23 [×2], C23 [×21], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×12], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×8], C2×D4 [×40], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×22], C24 [×3], C22⋊C8 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×6], C2×SD16 [×4], C2×Q16 [×2], C8.C22 [×4], C22×D4, C22×D4 [×2], C22×D4 [×3], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4, 2+ 1+4 [×8], (C22×C8)⋊C2, C23.24D4, C23.37D4, C22⋊SD16 [×4], D4.7D4 [×4], C23.38C23, C22×SD16, C2×C8.C22, C2×2+ 1+4, D4.(C2×D4)
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○SD16 [×2], D4.(C2×D4)

Smallest permutation representation of D4.(C2×D4)
On 32 points
Generators in S32
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)
(2 29)(3 7)(4 27)(6 25)(8 31)(10 20)(11 15)(12 18)(14 24)(16 22)(19 23)(28 32)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)
(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 29 5 25)(2 32 6 28)(3 27 7 31)(4 30 8 26)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 ··· 4 2 2 2 2 4 4 4 4 8 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○SD16 kernel D4.(C2×D4) (C22×C8)⋊C2 C23.24D4 C23.37D4 C22⋊SD16 D4.7D4 C23.38C23 C22×SD16 C2×C8.C22 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 1 1 4 4 1 1 1 1 7 1 4 4

Matrix representation of D4.(C2×D4) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 15 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 16 16 16 0
,
 1 0 0 0 0 0 7 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 16 0 16 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 15 0 0 16 0 16 1 0 0 0 1 0 1 0 0 1 0 0 16
,
 10 2 0 0 0 0 10 7 0 0 0 0 0 0 10 10 0 0 0 0 12 0 0 0 0 0 0 12 5 12 0 0 5 5 5 5
,
 10 2 0 0 0 0 10 7 0 0 0 0 0 0 10 10 0 0 0 0 12 7 0 0 0 0 0 12 5 12 0 0 5 5 12 12

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,15,1,1,16],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,0,12,5,0,0,0,0,5,5,0,0,0,0,12,5],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,7,12,5,0,0,0,0,5,12,0,0,0,0,12,12] >;

D4.(C2×D4) in GAP, Magma, Sage, TeX

D_4.(C_2\times D_4)
% in TeX

G:=Group("D4.(C2xD4)");
// GroupNames label

G:=SmallGroup(128,1741);
// by ID

G=gap.SmallGroup(128,1741);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,521,2804,1411,172]);
// Polycyclic

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

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