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G = C4×C22.D4order 128 = 27

Direct product of C4 and C22.D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C22.D4
G = < a,b,c,d,e | a4=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 556 in 344 conjugacy classes, 160 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22.D4, C23×C4, C22×D4, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.34D4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C22×C42, C2×C4×D4, C2×C22.D4, C4×C22.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C22.D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C2×C22.D4, C22.19C24, C23.36C23, C4×C22.D4

Smallest permutation representation of C4×C22.D4
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 44)(2 41)(3 42)(4 43)(5 46)(6 47)(7 48)(8 45)(9 36)(10 33)(11 34)(12 35)(13 26)(14 27)(15 28)(16 25)(17 62)(18 63)(19 64)(20 61)(21 53)(22 54)(23 55)(24 56)(29 52)(30 49)(31 50)(32 51)(37 60)(38 57)(39 58)(40 59)
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 61)(14 62)(15 63)(16 64)(17 27)(18 28)(19 25)(20 26)(29 39)(30 40)(31 37)(32 38)(33 43)(34 44)(35 41)(36 42)(45 55)(46 56)(47 53)(48 54)(49 59)(50 60)(51 57)(52 58)
(1 31 8 26)(2 32 5 27)(3 29 6 28)(4 30 7 25)(9 39 21 18)(10 40 22 19)(11 37 23 20)(12 38 24 17)(13 34 50 55)(14 35 51 56)(15 36 52 53)(16 33 49 54)(41 57 46 62)(42 58 47 63)(43 59 48 64)(44 60 45 61)
(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), (1,44)(2,41)(3,42)(4,43)(5,46)(6,47)(7,48)(8,45)(9,36)(10,33)(11,34)(12,35)(13,26)(14,27)(15,28)(16,25)(17,62)(18,63)(19,64)(20,61)(21,53)(22,54)(23,55)(24,56)(29,52)(30,49)(31,50)(32,51)(37,60)(38,57)(39,58)(40,59), (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,61)(14,62)(15,63)(16,64)(17,27)(18,28)(19,25)(20,26)(29,39)(30,40)(31,37)(32,38)(33,43)(34,44)(35,41)(36,42)(45,55)(46,56)(47,53)(48,54)(49,59)(50,60)(51,57)(52,58), (1,31,8,26)(2,32,5,27)(3,29,6,28)(4,30,7,25)(9,39,21,18)(10,40,22,19)(11,37,23,20)(12,38,24,17)(13,34,50,55)(14,35,51,56)(15,36,52,53)(16,33,49,54)(41,57,46,62)(42,58,47,63)(43,59,48,64)(44,60,45,61), (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), (1,44)(2,41)(3,42)(4,43)(5,46)(6,47)(7,48)(8,45)(9,36)(10,33)(11,34)(12,35)(13,26)(14,27)(15,28)(16,25)(17,62)(18,63)(19,64)(20,61)(21,53)(22,54)(23,55)(24,56)(29,52)(30,49)(31,50)(32,51)(37,60)(38,57)(39,58)(40,59), (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,61)(14,62)(15,63)(16,64)(17,27)(18,28)(19,25)(20,26)(29,39)(30,40)(31,37)(32,38)(33,43)(34,44)(35,41)(36,42)(45,55)(46,56)(47,53)(48,54)(49,59)(50,60)(51,57)(52,58), (1,31,8,26)(2,32,5,27)(3,29,6,28)(4,30,7,25)(9,39,21,18)(10,40,22,19)(11,37,23,20)(12,38,24,17)(13,34,50,55)(14,35,51,56)(15,36,52,53)(16,33,49,54)(41,57,46,62)(42,58,47,63)(43,59,48,64)(44,60,45,61), (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)], [(1,44),(2,41),(3,42),(4,43),(5,46),(6,47),(7,48),(8,45),(9,36),(10,33),(11,34),(12,35),(13,26),(14,27),(15,28),(16,25),(17,62),(18,63),(19,64),(20,61),(21,53),(22,54),(23,55),(24,56),(29,52),(30,49),(31,50),(32,51),(37,60),(38,57),(39,58),(40,59)], [(1,11),(2,12),(3,9),(4,10),(5,24),(6,21),(7,22),(8,23),(13,61),(14,62),(15,63),(16,64),(17,27),(18,28),(19,25),(20,26),(29,39),(30,40),(31,37),(32,38),(33,43),(34,44),(35,41),(36,42),(45,55),(46,56),(47,53),(48,54),(49,59),(50,60),(51,57),(52,58)], [(1,31,8,26),(2,32,5,27),(3,29,6,28),(4,30,7,25),(9,39,21,18),(10,40,22,19),(11,37,23,20),(12,38,24,17),(13,34,50,55),(14,35,51,56),(15,36,52,53),(16,33,49,54),(41,57,46,62),(42,58,47,63),(43,59,48,64),(44,60,45,61)], [(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 2J 2K 2L 2M 4A ··· 4H 4I ··· 4AB 4AC ··· 4AP order 1 2 ··· 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 C4○D4 kernel C4×C22.D4 C4×C22⋊C4 C4×C4⋊C4 C23.34D4 C23.8Q8 C23.23D4 C23.63C23 C24.C22 C22×C42 C2×C4×D4 C2×C22.D4 C22.D4 C22×C4 C2×C4 C23 # reps 1 3 2 1 1 1 2 2 1 1 1 16 4 16 4

Matrix representation of C4×C22.D4 in GL5(𝔽5)

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

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

C4×C22.D4 in GAP, Magma, Sage, TeX

C_4\times C_2^2.D_4
% in TeX

G:=Group("C4xC2^2.D4");
// GroupNames label

G:=SmallGroup(128,1033);
// by ID

G=gap.SmallGroup(128,1033);
# by ID

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

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

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