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## G = C2×C42.C22order 128 = 27

### Direct product of C2 and C42.C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4.4D4 — C2×C42.C22
 Lower central C1 — C22 — C2×C4 — C2×C42.C22
 Upper central C1 — C23 — C2×C42 — C2×C42.C22
 Jennings C1 — C22 — C22 — C42 — C2×C42.C22

Generators and relations for C2×C42.C22
G = < a,b,c,d,e | a2=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe=b-1, cd=dc, ece=b2c-1, ede=b-1c2d >

Subgroups: 324 in 142 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×8], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×8], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×3], C24, C8⋊C4 [×4], C8⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×C8 [×2], C22×D4, C22×Q8, C42.C22 [×4], C2×C8⋊C4 [×2], C2×C4.4D4, C2×C42.C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C42.C22 [×4], C2×C4.D4, C2×C4≀C2 [×2], C2×C42.C22

Smallest permutation representation of C2×C42.C22
On 64 points
Generators in S64
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(1 54 63 23)(2 51 64 20)(3 56 57 17)(4 53 58 22)(5 50 59 19)(6 55 60 24)(7 52 61 21)(8 49 62 18)(9 41 40 25)(10 46 33 30)(11 43 34 27)(12 48 35 32)(13 45 36 29)(14 42 37 26)(15 47 38 31)(16 44 39 28)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(2 35)(3 45)(4 10)(5 27)(6 39)(7 41)(8 14)(9 52)(11 19)(12 64)(13 56)(15 23)(16 60)(17 36)(18 42)(20 32)(21 40)(22 46)(24 28)(25 61)(26 49)(29 57)(30 53)(33 58)(34 50)(37 62)(38 54)(43 59)(44 55)(47 63)(48 51)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 D4 C4≀C2 C4.D4 kernel C2×C42.C22 C42.C22 C2×C8⋊C4 C2×C4.4D4 C4.4D4 C22×D4 C22×Q8 C42 C22×C4 C22 C22 # reps 1 4 2 1 4 2 2 2 2 16 2

Matrix representation of C2×C42.C22 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
,
 1 0 0 0 0 0 0 13 0 0 0 13 0 0 0 0 0 0 1 15 0 0 0 1 16
,
 16 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 4
,
 13 0 0 0 0 0 6 10 0 0 0 10 6 0 0 0 0 0 0 12 0 0 0 6 0
,
 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 16 1

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],[1,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,1,1,0,0,0,15,16],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[13,0,0,0,0,0,6,10,0,0,0,10,6,0,0,0,0,0,0,6,0,0,0,12,0],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,16,0,0,0,0,1] >;

C2×C42.C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2.C_2^2
% in TeX

G:=Group("C2xC4^2.C2^2");
// GroupNames label

G:=SmallGroup(128,254);
// by ID

G=gap.SmallGroup(128,254);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,1018,248,1971,102]);
// Polycyclic

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

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