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## G = C22×C4⋊D4order 128 = 27

### Direct product of C22 and C4⋊D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C4⋊D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C22×C4⋊D4
 Lower central C1 — C22 — C22×C4⋊D4
 Upper central C1 — C24 — C22×C4⋊D4
 Jennings C1 — C22 — C22×C4⋊D4

Generators and relations for C22×C4⋊D4
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, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 2188 in 1264 conjugacy classes, 516 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C23×C4, C23×C4, C22×D4, C22×D4, C25, C25, C22×C22⋊C4, C22×C4⋊C4, C2×C4⋊D4, C24×C4, D4×C23, D4×C23, C22×C4⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C25, C2×C4⋊D4, D4×C23, C22×C4○D4, C22×C4⋊D4

Smallest permutation representation of C22×C4⋊D4
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 23)(6 24)(7 21)(8 22)(9 59)(10 60)(11 57)(12 58)(13 19)(14 20)(15 17)(16 18)(25 42)(26 43)(27 44)(28 41)(33 56)(34 53)(35 54)(36 55)(37 63)(38 64)(39 61)(40 62)(45 51)(46 52)(47 49)(48 50)
(1 47)(2 48)(3 45)(4 46)(5 9)(6 10)(7 11)(8 12)(13 36)(14 33)(15 34)(16 35)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 37)(26 38)(27 39)(28 40)(29 49)(30 50)(31 51)(32 52)(41 62)(42 63)(43 64)(44 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 27 55 59)(2 26 56 58)(3 25 53 57)(4 28 54 60)(5 49 61 13)(6 52 62 16)(7 51 63 15)(8 50 64 14)(9 29 44 36)(10 32 41 35)(11 31 42 34)(12 30 43 33)(17 21 45 37)(18 24 46 40)(19 23 47 39)(20 22 48 38)
(2 4)(5 61)(6 64)(7 63)(8 62)(9 44)(10 43)(11 42)(12 41)(14 16)(18 20)(21 37)(22 40)(23 39)(24 38)(25 57)(26 60)(27 59)(28 58)(30 32)(33 35)(46 48)(50 52)(54 56)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 kernel C22×C4⋊D4 C22×C22⋊C4 C22×C4⋊C4 C2×C4⋊D4 C24×C4 D4×C23 C22×C4 C24 C23 # reps 1 2 1 24 1 3 8 8 8

Matrix representation of C22×C4⋊D4 in GL7(𝔽5)

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,1,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,0,4,2,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,2,4,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,4] >;

C22×C4⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes D_4
% in TeX

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

G:=SmallGroup(128,2164);
// by ID

G=gap.SmallGroup(128,2164);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430]);
// 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,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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