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G = C23×C22⋊C4  order 128 = 27

Direct product of C23 and C22⋊C4

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

Aliases: C23×C22⋊C4, C25⋊5C4, C26.2C2, C24.193D4, C22.5C25, C25.92C22, C23.101C24, C24.601C23, (C2×C4)⋊3C24, (C24×C4)⋊4C2, C24⋊18(C2×C4), C2.1(C24×C4), C2.1(D4×C23), C22⋊2(C23×C4), C23⋊7(C22×C4), (C22×C4)⋊21C23, (C23×C4)⋊56C22, C23.886(C2×D4), C22.154(C22×D4), SmallGroup(128,2151)

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

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

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
010000
004000
000400
000040
000004
,
100000
040000
001000
000400
000010
000001
,
100000
040000
004000
000100
000040
000004
,
100000
040000
004000
000100
000010
000004
,
100000
010000
001000
000100
000040
000004
,
200000
010000
002000
000400
000001
000040

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] >;
 

80 conjugacy classes

class 1 2A···2AE2AF···2AU4A···4AF
order12···22···24···4
size11···12···22···2

80 irreducible representations

dim111112
type+++++
imageC1C2C2C2C4D4
kernelC23×C22⋊C4C22×C22⋊C4C24×C4C26C25C24
# reps128213216

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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