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## G = (C2×D4).301D4order 128 = 27

### 54th non-split extension by C2×D4 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×D4).301D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.33C23 — (C2×D4).301D4
 Lower central C1 — C2 — C2×C4 — (C2×D4).301D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×D4).301D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×D4).301D4

Generators and relations for (C2×D4).301D4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe=ab-1, dcd-1=ece=ab2c, ede=ad3 >

Subgroups: 452 in 209 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×13], D4 [×16], Q8 [×2], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×4], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C22≀C2 [×2], C4⋊D4 [×4], C4.4D4, C41D4, C22×C8, C2×M4(2), C22×D4, C2×C4○D4, (C22×C8)⋊C2, C2×D4⋊C4, C23.37D4, C23.25D4, M4(2)⋊C4, C22.D8 [×2], C23.46D4 [×2], C23.19D4 [×4], C23.33C23, C22.29C24, (C2×D4).301D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○D8, D4○SD16, (C2×D4).301D4

Smallest permutation representation of (C2×D4).301D4
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(1 32 5 28)(2 15 6 11)(3 26 7 30)(4 9 8 13)(10 23 14 19)(12 17 16 21)(18 27 22 31)(20 29 24 25)
(1 28)(2 11)(3 30)(4 13)(5 32)(6 15)(7 26)(8 9)(10 23)(12 17)(14 19)(16 21)(18 31)(20 25)(22 27)(24 29)
(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 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 28)(12 26)(13 15)(14 32)(16 30)(17 21)(25 31)(27 29)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D4○D8 D4○SD16 kernel (C2×D4).301D4 (C22×C8)⋊C2 C2×D4⋊C4 C23.37D4 C23.25D4 M4(2)⋊C4 C22.D8 C23.46D4 C23.19D4 C23.33C23 C22.29C24 C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 1 1 1 1 1 2 2 4 1 1 3 1 8 2 2

Matrix representation of (C2×D4).301D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 1 1 0 0 1 0 16 0 0 0 16 16 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 1 1 0 0 0 0 16 0 0 0 0 1 1 0
,
 13 0 0 0 0 0 0 4 0 0 0 0 0 0 6 6 0 0 0 0 14 0 0 0 0 0 0 3 3 14 0 0 14 14 3 3
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 16 0 0 16

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

(C2×D4).301D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{301}D_4
% in TeX

G:=Group("(C2xD4).301D4");
// GroupNames label

G:=SmallGroup(128,1828);
// by ID

G=gap.SmallGroup(128,1828);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,4037,1027,124]);
// Polycyclic

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

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