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## G = C22×C4.D4order 128 = 27

### Direct product of C22 and C4.D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22×C4.D4
G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 1068 in 496 conjugacy classes, 180 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C22, C22 [×10], C22 [×76], C8 [×8], C2×C4, C2×C4 [×27], D4 [×32], C23, C23 [×14], C23 [×84], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×14], C2×D4 [×16], C2×D4 [×48], C24, C24 [×12], C24 [×16], C4.D4 [×16], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C22×D4 [×12], C22×D4 [×8], C25 [×2], C2×C4.D4 [×12], C22×M4(2) [×2], D4×C23, C22×C4.D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C4.D4 [×4], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4.D4 [×6], C22×C22⋊C4, C22×C4.D4

Smallest permutation representation of C22×C4.D4
On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 24 13 20)(10 21 14 17)(11 18 15 22)(12 23 16 19)
(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 30 29 2 5 26 25 6)(3 28 31 8 7 32 27 4)(9 12 24 23 13 16 20 19)(10 18 21 15 14 22 17 11)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 4A ··· 4H 8A ··· 8P order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 C4 C4 D4 C4.D4 kernel C22×C4.D4 C2×C4.D4 C22×M4(2) D4×C23 C22×D4 C25 C22×C4 C22 # reps 1 12 2 1 12 4 8 4

Matrix representation of C22×C4.D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 13 0 1 0 0 0 0 4 4 16 0
,
 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 13 0 0 15 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 13 0 0 0 0 0 1 0 4
,
 16 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 4 0 15 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 13 0 0 0 0 0 1 0 4 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,4,0,0,0,0,15,1,13,4,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,15,1,13,4],[16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,1,13,4,0,0,0,0,0,1,0,0] >;

C22×C4.D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4.D_4
% in TeX

G:=Group("C2^2xC4.D4");
// GroupNames label

G:=SmallGroup(128,1617);
// by ID

G=gap.SmallGroup(128,1617);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,2028,124]);
// Polycyclic

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

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