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## G = C2×C22.C42order 128 = 27

### Direct product of C2 and C22.C42

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C22.C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 356 in 216 conjugacy classes, 116 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22.C42, C22×C4⋊C4, C22×M4(2), C2×C22.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C4.D4, C4.10D4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22.C42, C2×C2.C42, C2×C4.D4, C2×C4.10D4, C2×C22.C42

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + - + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 C4.D4 C4.10D4 kernel C2×C22.C42 C22.C42 C22×C4⋊C4 C22×M4(2) C2×C4⋊C4 C2×M4(2) C23×C4 C22×C4 C22×C4 C22 C22 # reps 1 4 1 2 4 16 4 6 2 2 2

Matrix representation of C2×C22.C42 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 11 16 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0
,
 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 15 9 0 0 0 0 0 0 11 2 0 0 0 0 0 0 0 0 2 8 0 0 0 0 0 0 6 15

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[11,1,0,0,0,0,0,0,16,6,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,15,11,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0,8,15] >;

C2×C22.C42 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC2^2.C4^2");
// GroupNames label

G:=SmallGroup(128,473);
// by ID

G=gap.SmallGroup(128,473);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,1411,172]);
// Polycyclic

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

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