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## G = C22×C4⋊1D4order 128 = 27

### Direct product of C22 and C4⋊1D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C4⋊1D4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C22×C4⋊1D4
 Lower central C1 — C22 — C22×C4⋊1D4
 Upper central C1 — C24 — C22×C4⋊1D4
 Jennings C1 — C22 — C22×C4⋊1D4

Generators and relations for C22×C41D4
G = < a,b,c,d,e | a2=b2=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2796 in 1500 conjugacy classes, 556 normal (5 characteristic)
C1, C2 [×15], C2 [×16], C4 [×24], C22, C22 [×34], C22 [×144], C2×C4 [×84], D4 [×192], C23 [×31], C23 [×208], C42 [×16], C22×C4 [×42], C2×D4 [×96], C2×D4 [×288], C24, C24 [×24], C24 [×48], C2×C42 [×12], C41D4 [×64], C23×C4 [×3], C22×D4 [×72], C22×D4 [×48], C25 [×4], C22×C42, C2×C41D4 [×24], D4×C23 [×6], C22×C41D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×24], C23 [×155], C2×D4 [×84], C24 [×31], C41D4 [×16], C22×D4 [×42], C25, C2×C41D4 [×12], D4×C23 [×3], C22×C41D4

Smallest permutation representation of C22×C41D4
On 64 points
Generators in S64
(1 24)(2 21)(3 22)(4 23)(5 56)(6 53)(7 54)(8 55)(9 48)(10 45)(11 46)(12 47)(13 64)(14 61)(15 62)(16 63)(17 35)(18 36)(19 33)(20 34)(25 38)(26 39)(27 40)(28 37)(29 43)(30 44)(31 41)(32 42)(49 60)(50 57)(51 58)(52 59)
(1 18)(2 19)(3 20)(4 17)(5 46)(6 47)(7 48)(8 45)(9 54)(10 55)(11 56)(12 53)(13 58)(14 59)(15 60)(16 57)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(29 40)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 61)
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 31 59 55)(2 32 60 56)(3 29 57 53)(4 30 58 54)(5 21 42 49)(6 22 43 50)(7 23 44 51)(8 24 41 52)(9 17 37 13)(10 18 38 14)(11 19 39 15)(12 20 40 16)(25 61 45 36)(26 62 46 33)(27 63 47 34)(28 64 48 35)
(1 50)(2 49)(3 52)(4 51)(5 56)(6 55)(7 54)(8 53)(9 48)(10 47)(11 46)(12 45)(13 35)(14 34)(15 33)(16 36)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 40)(26 39)(27 38)(28 37)(29 41)(30 44)(31 43)(32 42)

G:=sub<Sym(64)| (1,24)(2,21)(3,22)(4,23)(5,56)(6,53)(7,54)(8,55)(9,48)(10,45)(11,46)(12,47)(13,64)(14,61)(15,62)(16,63)(17,35)(18,36)(19,33)(20,34)(25,38)(26,39)(27,40)(28,37)(29,43)(30,44)(31,41)(32,42)(49,60)(50,57)(51,58)(52,59), (1,18)(2,19)(3,20)(4,17)(5,46)(6,47)(7,48)(8,45)(9,54)(10,55)(11,56)(12,53)(13,58)(14,59)(15,60)(16,57)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(29,40)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,61), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,31,59,55)(2,32,60,56)(3,29,57,53)(4,30,58,54)(5,21,42,49)(6,22,43,50)(7,23,44,51)(8,24,41,52)(9,17,37,13)(10,18,38,14)(11,19,39,15)(12,20,40,16)(25,61,45,36)(26,62,46,33)(27,63,47,34)(28,64,48,35), (1,50)(2,49)(3,52)(4,51)(5,56)(6,55)(7,54)(8,53)(9,48)(10,47)(11,46)(12,45)(13,35)(14,34)(15,33)(16,36)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,40)(26,39)(27,38)(28,37)(29,41)(30,44)(31,43)(32,42)>;

G:=Group( (1,24)(2,21)(3,22)(4,23)(5,56)(6,53)(7,54)(8,55)(9,48)(10,45)(11,46)(12,47)(13,64)(14,61)(15,62)(16,63)(17,35)(18,36)(19,33)(20,34)(25,38)(26,39)(27,40)(28,37)(29,43)(30,44)(31,41)(32,42)(49,60)(50,57)(51,58)(52,59), (1,18)(2,19)(3,20)(4,17)(5,46)(6,47)(7,48)(8,45)(9,54)(10,55)(11,56)(12,53)(13,58)(14,59)(15,60)(16,57)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(29,40)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,61), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,31,59,55)(2,32,60,56)(3,29,57,53)(4,30,58,54)(5,21,42,49)(6,22,43,50)(7,23,44,51)(8,24,41,52)(9,17,37,13)(10,18,38,14)(11,19,39,15)(12,20,40,16)(25,61,45,36)(26,62,46,33)(27,63,47,34)(28,64,48,35), (1,50)(2,49)(3,52)(4,51)(5,56)(6,55)(7,54)(8,53)(9,48)(10,47)(11,46)(12,45)(13,35)(14,34)(15,33)(16,36)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,40)(26,39)(27,38)(28,37)(29,41)(30,44)(31,43)(32,42) );

G=PermutationGroup([(1,24),(2,21),(3,22),(4,23),(5,56),(6,53),(7,54),(8,55),(9,48),(10,45),(11,46),(12,47),(13,64),(14,61),(15,62),(16,63),(17,35),(18,36),(19,33),(20,34),(25,38),(26,39),(27,40),(28,37),(29,43),(30,44),(31,41),(32,42),(49,60),(50,57),(51,58),(52,59)], [(1,18),(2,19),(3,20),(4,17),(5,46),(6,47),(7,48),(8,45),(9,54),(10,55),(11,56),(12,53),(13,58),(14,59),(15,60),(16,57),(21,33),(22,34),(23,35),(24,36),(25,41),(26,42),(27,43),(28,44),(29,40),(30,37),(31,38),(32,39),(49,62),(50,63),(51,64),(52,61)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,31,59,55),(2,32,60,56),(3,29,57,53),(4,30,58,54),(5,21,42,49),(6,22,43,50),(7,23,44,51),(8,24,41,52),(9,17,37,13),(10,18,38,14),(11,19,39,15),(12,20,40,16),(25,61,45,36),(26,62,46,33),(27,63,47,34),(28,64,48,35)], [(1,50),(2,49),(3,52),(4,51),(5,56),(6,55),(7,54),(8,53),(9,48),(10,47),(11,46),(12,45),(13,35),(14,34),(15,33),(16,36),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,40),(26,39),(27,38),(28,37),(29,41),(30,44),(31,43),(32,42)])

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 4A ··· 4X order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 D4 kernel C22×C4⋊1D4 C22×C42 C2×C4⋊1D4 D4×C23 C22×C4 # reps 1 1 24 6 24

Matrix representation of C22×C41D4 in GL6(ℤ)

 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 2 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 2 -1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,2,0,0,0,0,-1,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,2,0,0,0,0,0,-1] >;

C22×C41D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes_1D_4
% in TeX

G:=Group("C2^2xC4:1D4");
// GroupNames label

G:=SmallGroup(128,2172);
// by ID

G=gap.SmallGroup(128,2172);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,352]);
// Polycyclic

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

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