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## G = C2×C22⋊D8order 128 = 27

### Direct product of C2 and C22⋊D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C22⋊D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C2×C22⋊D8
 Lower central C1 — C2 — C2×C4 — C2×C22⋊D8
 Upper central C1 — C23 — C23×C4 — C2×C22⋊D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C22⋊D8

Generators and relations for C2×C22⋊D8
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1276 in 524 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×14], C4 [×4], C4 [×4], C22, C22 [×10], C22 [×86], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×14], D4 [×8], D4 [×42], C23, C23 [×6], C23 [×96], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], D8 [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×5], C2×D4 [×14], C2×D4 [×63], C24, C24 [×23], C22⋊C8 [×4], D4⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C2×D8 [×8], C2×D8 [×8], C23×C4, C22×D4, C22×D4 [×6], C22×D4 [×12], C25, C2×C22⋊C8, C2×D4⋊C4 [×2], C22⋊D8 [×8], C2×C4⋊D4, C22×D8 [×2], D4×C23, C2×C22⋊D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D8 [×4], C2×D4 [×18], C24, C22≀C2 [×4], C2×D8 [×6], C8⋊C22 [×2], C22×D4 [×3], C22⋊D8 [×4], C2×C22≀C2, C22×D8, C2×C8⋊C22, C2×C22⋊D8

Smallest permutation representation of C2×C22⋊D8
On 32 points
Generators in S32
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(2 31)(4 25)(6 27)(8 29)(9 18)(11 20)(13 22)(15 24)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 32)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D8 C8⋊C22 kernel C2×C22⋊D8 C2×C22⋊C8 C2×D4⋊C4 C22⋊D8 C2×C4⋊D4 C22×D8 D4×C23 C22×C4 C2×D4 C24 C23 C22 # reps 1 1 2 8 1 2 1 3 8 1 8 2

Matrix representation of C2×C22⋊D8 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 3 14 0 0 0 3 3 0 0 0 0 0 1 15 0 0 0 1 16
,
 16 0 0 0 0 0 3 3 0 0 0 3 14 0 0 0 0 0 1 15 0 0 0 0 16

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,1,1,0,0,0,15,16],[16,0,0,0,0,0,3,3,0,0,0,3,14,0,0,0,0,0,1,0,0,0,0,15,16] >;

C2×C22⋊D8 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes D_8
% in TeX

G:=Group("C2xC2^2:D8");
// GroupNames label

G:=SmallGroup(128,1728);
// by ID

G=gap.SmallGroup(128,1728);
# by ID

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

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

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