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## G = (C2×D4)⋊21D4order 128 = 27

### 17th semidirect product of C2×D4 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×D4)⋊21D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — (C2×D4)⋊21D4
 Lower central C1 — C2 — C2×C4 — (C2×D4)⋊21D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×D4)⋊21D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×D4)⋊21D4

Generators and relations for (C2×D4)⋊21D4
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, eae=ab2, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=d-1 >

Subgroups: 836 in 382 conjugacy classes, 108 normal (38 characteristic)
C1, C2 [×3], C2 [×11], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×35], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×23], D4 [×6], D4 [×39], Q8 [×2], Q8 [×5], C23, C23 [×2], C23 [×22], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×6], SD16 [×8], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4 [×3], C2×D4 [×8], C2×D4 [×43], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×26], C24 [×3], C22⋊C8 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×2], C2×Q16, C4○D8 [×4], C8⋊C22 [×4], C22×D4, C22×D4 [×2], C22×D4 [×3], C2×C4○D4 [×3], C2×C4○D4 [×2], C2×C4○D4, 2+ 1+4 [×8], (C22×C8)⋊C2, C2×D4⋊C4, C23.36D4, C22⋊D8 [×2], D4⋊D4 [×2], C22⋊SD16 [×2], D4.7D4 [×2], C22.31C24, C2×C4○D8, C2×C8⋊C22, C2×2+ 1+4, (C2×D4)⋊21D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○D8, D4○SD16, (C2×D4)⋊21D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 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 C2 D4 D4 D4 D4○D8 D4○SD16 kernel (C2×D4)⋊21D4 (C22×C8)⋊C2 C2×D4⋊C4 C23.36D4 C22⋊D8 D4⋊D4 C22⋊SD16 D4.7D4 C22.31C24 C2×C4○D8 C2×C8⋊C22 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 7 1 4 2 2

Matrix representation of (C2×D4)⋊21D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 16 0 1 0 0 0 0 16 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 1 0 0 0 0 0 12 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 16 0 1 0 0 0 0 1 0 16
,
 10 4 0 0 0 0 13 7 0 0 0 0 0 0 3 14 0 0 0 0 14 14 0 0 0 0 0 0 3 14 0 0 0 0 14 14
,
 7 13 0 0 0 0 12 10 0 0 0 0 0 0 3 14 11 6 0 0 14 14 6 6 0 0 0 0 14 3 0 0 0 0 3 3

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,12,0,0,0,0,0,16,0,0,0,0,0,0,16,0,16,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,16],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[7,12,0,0,0,0,13,10,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,11,6,14,3,0,0,6,6,3,3] >;

(C2×D4)⋊21D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_{21}D_4
% in TeX

G:=Group("(C2xD4):21D4");
// GroupNames label

G:=SmallGroup(128,1744);
// by ID

G=gap.SmallGroup(128,1744);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,248,2804,1411,172]);
// Polycyclic

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

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