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## G = C22×C42⋊2C2order 128 = 27

### Direct product of C22 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C42⋊2C2
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C22×C42⋊2C2
 Lower central C1 — C22 — C22×C42⋊2C2
 Upper central C1 — C24 — C22×C42⋊2C2
 Jennings C1 — C22 — C22×C42⋊2C2

Generators and relations for C22×C422C2
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, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 972 in 660 conjugacy classes, 436 normal (6 characteristic)
C1, C2 [×15], C2 [×4], C4 [×24], C22, C22 [×34], C22 [×36], C2×C4 [×24], C2×C4 [×72], C23 [×19], C23 [×52], C42 [×16], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×36], C22×C4 [×24], C24, C24 [×6], C24 [×12], C2×C42 [×12], C2×C22⋊C4 [×36], C2×C4⋊C4 [×36], C422C2 [×64], C23×C4 [×6], C25, C22×C42, C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C2×C422C2 [×24], C22×C422C2
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×12], C24 [×31], C422C2 [×16], C2×C4○D4 [×18], C25, C2×C422C2 [×12], C22×C4○D4 [×3], C22×C422C2

Smallest permutation representation of C22×C422C2
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 39)(6 40)(7 37)(8 38)(9 34)(10 35)(11 36)(12 33)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 63)(22 64)(23 61)(24 62)(25 55)(26 56)(27 53)(28 54)(41 59)(42 60)(43 57)(44 58)
(1 19)(2 20)(3 17)(4 18)(5 53)(6 54)(7 55)(8 56)(9 45)(10 46)(11 47)(12 48)(13 36)(14 33)(15 34)(16 35)(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 61)(42 62)(43 63)(44 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 25 9 59)(2 26 10 60)(3 27 11 57)(4 28 12 58)(5 13 63 51)(6 14 64 52)(7 15 61 49)(8 16 62 50)(17 39 47 21)(18 40 48 22)(19 37 45 23)(20 38 46 24)(29 55 34 41)(30 56 35 42)(31 53 36 43)(32 54 33 44)
(1 51)(2 14)(3 49)(4 16)(5 57)(6 28)(7 59)(8 26)(9 13)(10 52)(11 15)(12 50)(17 29)(18 35)(19 31)(20 33)(21 53)(22 44)(23 55)(24 42)(25 61)(27 63)(30 48)(32 46)(34 47)(36 45)(37 41)(38 56)(39 43)(40 54)(58 64)(60 62)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P 2Q 2R 2S 4A ··· 4X 4Y ··· 4AJ order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C2 C4○D4 kernel C22×C42⋊2C2 C22×C42 C22×C22⋊C4 C22×C4⋊C4 C2×C42⋊2C2 C23 # reps 1 1 3 3 24 24

Matrix representation of C22×C422C2 in GL8(𝔽5)

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

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2],[4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1] >;

C22×C422C2 in GAP, Magma, Sage, TeX

C_2^2\times C_4^2\rtimes_2C_2
% in TeX

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

G:=SmallGroup(128,2170);
// by ID

G=gap.SmallGroup(128,2170);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,184]);
// 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,c*d=d*c,e*c*e=c*d^2,e*d*e=c^2*d^-1>;
// generators/relations

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