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## G = C4×C42⋊2C2order 128 = 27

### Direct product of C4 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C42⋊2C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C42⋊2C2
 Lower central C1 — C22 — C4×C42⋊2C2
 Upper central C1 — C22×C4 — C4×C42⋊2C2
 Jennings C1 — C23 — C4×C42⋊2C2

Generators and relations for C4×C422C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 396 in 258 conjugacy classes, 148 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×22], C22, C22 [×6], C22 [×10], C2×C4 [×24], C2×C4 [×34], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×18], C22⋊C4 [×12], C22⋊C4 [×6], C4⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×12], C22×C4 [×6], C24, C2.C42 [×12], C2×C42, C2×C42 [×9], C2×C22⋊C4 [×6], C2×C4⋊C4 [×6], C422C2 [×8], C23×C4, C43, C4×C22⋊C4 [×3], C4×C4⋊C4 [×3], C425C4, C23.63C23 [×3], C24.C22 [×3], C2×C422C2, C4×C422C2
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×12], C24, C422C2 [×4], C23×C4, C2×C4○D4 [×6], C4×C4○D4 [×3], C2×C422C2, C23.36C23 [×3], C4×C422C2

Smallest permutation representation of C4×C422C2
On 64 points
Generators in S64
(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 34 8 55)(2 35 5 56)(3 36 6 53)(4 33 7 54)(9 42 21 47)(10 43 22 48)(11 44 23 45)(12 41 24 46)(13 37 50 20)(14 38 51 17)(15 39 52 18)(16 40 49 19)(25 64 30 59)(26 61 31 60)(27 62 32 57)(28 63 29 58)
(1 29 11 39)(2 30 12 40)(3 31 9 37)(4 32 10 38)(5 25 24 19)(6 26 21 20)(7 27 22 17)(8 28 23 18)(13 53 61 47)(14 54 62 48)(15 55 63 45)(16 56 64 46)(33 57 43 51)(34 58 44 52)(35 59 41 49)(36 60 42 50)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 52)(14 49)(15 50)(16 51)(17 30)(18 31)(19 32)(20 29)(21 23)(22 24)(25 38)(26 39)(27 40)(28 37)(33 41)(34 42)(35 43)(36 44)(45 53)(46 54)(47 55)(48 56)(57 64)(58 61)(59 62)(60 63)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4AF 4AG ··· 4AT order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4○D4 kernel C4×C42⋊2C2 C43 C4×C22⋊C4 C4×C4⋊C4 C42⋊5C4 C23.63C23 C24.C22 C2×C42⋊2C2 C42⋊2C2 C2×C4 # reps 1 1 3 3 1 3 3 1 16 24

Matrix representation of C4×C422C2 in GL5(𝔽5)

 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 2
,
 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 4 3 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 4 4

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,3,1],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,0,4] >;

C4×C422C2 in GAP, Magma, Sage, TeX

C_4\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C4xC4^2:2C2");
// GroupNames label

G:=SmallGroup(128,1036);
// by ID

G=gap.SmallGroup(128,1036);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,100,248]);
// Polycyclic

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

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