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## G = C2×C4×C22⋊C4order 128 = 27

### Direct product of C2×C4 and C22⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C4×C22⋊C4
G = < a,b,c,d,e | a2=b4=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 1020 in 680 conjugacy classes, 340 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C23×C4, C23×C4, C23×C4, C25, C2×C2.C42, C4×C22⋊C4, C22×C42, C22×C22⋊C4, C24×C4, C2×C4×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×C22⋊C4, C22×C42, C22×C22⋊C4, C2×C42⋊C2, C2×C4×D4, C2×C4×C22⋊C4

Smallest permutation representation of C2×C4×C22⋊C4
On 64 points
Generators in S64
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 43)(14 44)(15 41)(16 42)(17 32)(18 29)(19 30)(20 31)(25 36)(26 33)(27 34)(28 35)(37 47)(38 48)(39 45)(40 46)(49 58)(50 59)(51 60)(52 57)(53 64)(54 61)(55 62)(56 63)
(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 21)(2 22)(3 23)(4 24)(5 11)(6 12)(7 9)(8 10)(13 31)(14 32)(15 29)(16 30)(17 44)(18 41)(19 42)(20 43)(25 36)(26 33)(27 34)(28 35)(37 47)(38 48)(39 45)(40 46)(49 58)(50 59)(51 60)(52 57)(53 64)(54 61)(55 62)(56 63)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 62)(26 63)(27 64)(28 61)(29 41)(30 42)(31 43)(32 44)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(1 36 29 58)(2 33 30 59)(3 34 31 60)(4 35 32 57)(5 25 18 49)(6 26 19 50)(7 27 20 51)(8 28 17 52)(9 53 43 40)(10 54 44 37)(11 55 41 38)(12 56 42 39)(13 46 23 64)(14 47 24 61)(15 48 21 62)(16 45 22 63)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 C4○D4 kernel C2×C4×C22⋊C4 C2×C2.C42 C4×C22⋊C4 C22×C42 C22×C22⋊C4 C24×C4 C2×C22⋊C4 C23×C4 C22×C4 C23 # reps 1 2 8 2 2 1 32 16 8 8

Matrix representation of C2×C4×C22⋊C4 in GL5(𝔽5)

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

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

C2×C4×C22⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,1000);
// by ID

G=gap.SmallGroup(128,1000);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,184]);
// Polycyclic

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

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