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## G = C23×C22⋊C4order 128 = 27

### Direct product of C23 and C22⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C23×C22⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C26 — C23×C22⋊C4
 Lower central C1 — C2 — C23×C22⋊C4
 Upper central C1 — C25 — C23×C22⋊C4
 Jennings C1 — C22 — C23×C22⋊C4

Generators and relations for C23×C22⋊C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 3644 in 2316 conjugacy classes, 988 normal (6 characteristic)
C1, C2, C2 [×30], C2 [×16], C4 [×16], C22, C22 [×170], C22 [×240], C2×C4 [×16], C2×C4 [×112], C23 [×275], C23 [×560], C22⋊C4 [×64], C22×C4 [×56], C22×C4 [×112], C24 [×171], C24 [×240], C2×C22⋊C4 [×112], C23×C4 [×28], C23×C4 [×16], C25, C25 [×30], C25 [×16], C22×C22⋊C4 [×28], C24×C4 [×2], C26, C23×C22⋊C4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], D4 [×16], C23 [×155], C22⋊C4 [×64], C22×C4 [×140], C2×D4 [×56], C24 [×31], C2×C22⋊C4 [×112], C23×C4 [×30], C22×D4 [×28], C25, C22×C22⋊C4 [×28], C24×C4, D4×C23 [×2], C23×C22⋊C4

Smallest permutation representation of C23×C22⋊C4
On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 35)(6 36)(7 33)(8 34)(13 37)(14 38)(15 39)(16 40)(17 45)(18 46)(19 47)(20 48)(21 55)(22 56)(23 53)(24 54)(25 60)(26 57)(27 58)(28 59)(29 63)(30 64)(31 61)(32 62)(41 49)(42 50)(43 51)(44 52)
(1 35)(2 36)(3 33)(4 34)(5 9)(6 10)(7 11)(8 12)(13 51)(14 52)(15 49)(16 50)(17 23)(18 24)(19 21)(20 22)(25 30)(26 31)(27 32)(28 29)(37 43)(38 44)(39 41)(40 42)(45 53)(46 54)(47 55)(48 56)(57 61)(58 62)(59 63)(60 64)
(1 23)(2 24)(3 21)(4 22)(5 45)(6 46)(7 47)(8 48)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 35)(18 36)(19 33)(20 34)(25 40)(26 37)(27 38)(28 39)(29 41)(30 42)(31 43)(32 44)(49 63)(50 64)(51 61)(52 62)
(1 19)(2 52)(3 17)(4 50)(5 55)(6 38)(7 53)(8 40)(9 47)(10 44)(11 45)(12 42)(13 63)(14 36)(15 61)(16 34)(18 58)(20 60)(21 35)(22 64)(23 33)(24 62)(25 48)(26 41)(27 46)(28 43)(29 37)(30 56)(31 39)(32 54)(49 57)(51 59)
(1 59)(2 60)(3 57)(4 58)(5 29)(6 30)(7 31)(8 32)(9 28)(10 25)(11 26)(12 27)(13 21)(14 22)(15 23)(16 24)(17 49)(18 50)(19 51)(20 52)(33 61)(34 62)(35 63)(36 64)(37 55)(38 56)(39 53)(40 54)(41 45)(42 46)(43 47)(44 48)
(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)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2AE 2AF ··· 2AU 4A ··· 4AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C4 D4 kernel C23×C22⋊C4 C22×C22⋊C4 C24×C4 C26 C25 C24 # reps 1 28 2 1 32 16

Matrix representation of C23×C22⋊C4 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C23×C22⋊C4 in GAP, Magma, Sage, TeX

C_2^3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C2^3xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,2151);
// by ID

G=gap.SmallGroup(128,2151);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477]);
// Polycyclic

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

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