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G = C2×C23.24D4  order 128 = 27

Direct product of C2 and C23.24D4

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

Aliases: C2×C23.24D4, C24.139D4, (C23×C8)⋊6C2, C4.3(C23×C4), C4⋊C4.340C23, (C2×C4).173C24, (C2×C8).467C23, (C22×C8)⋊63C22, D4.17(C22×C4), (C22×C4).819D4, C4.138(C22×D4), C23.376(C2×D4), Q8.17(C22×C4), D4⋊C4⋊97C22, Q8⋊C4⋊99C22, (C2×D4).358C23, C22.81(C4○D8), C4○(C23.24D4), (C2×Q8).331C23, C42⋊C2⋊73C22, (C23×C4).690C22, C22.123(C22×D4), C23.130(C22⋊C4), (C22×C4).1497C23, (C22×D4).551C22, (C22×Q8).455C22, C4○(C2×D4⋊C4), C4○(C2×Q8⋊C4), C2.1(C2×C4○D8), (C2×C4○D4)⋊18C4, C4○D4⋊12(C2×C4), (C2×C4)○2(D4⋊C4), (C2×D4⋊C4)⋊58C2, (C2×C4)○2(Q8⋊C4), (C2×Q8⋊C4)⋊59C2, (C2×D4).226(C2×C4), (C2×C4).1563(C2×D4), C4.121(C2×C22⋊C4), (C2×Q8).204(C2×C4), (C2×C42⋊C2)⋊40C2, (C22×C4)○(D4⋊C4), (C2×C4⋊C4).899C22, (C22×C4).415(C2×C4), (C2×C4).458(C22×C4), (C22×C4)○(Q8⋊C4), (C22×C4○D4).18C2, C22.20(C2×C22⋊C4), C2.35(C22×C22⋊C4), (C2×C4).284(C22⋊C4), (C2×C4○D4).272C22, (C2×C4)○(C23.24D4), (C2×C4)○(C2×D4⋊C4), (C2×C4)○(C2×Q8⋊C4), (C22×C4)○(C2×D4⋊C4), (C22×C4)○(C2×Q8⋊C4), SmallGroup(128,1624)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×C23.24D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×C23.24D4
Lower central
C1 — C2 — C4 — C2×C23.24D4
Upper central
C1 — C22×C4 — C23×C4 — C2×C23.24D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C23.24D4

Subgroups: 668 in 396 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×28], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×30], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×14], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×15], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×6], C22×C8 [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C23.24D4 [×8], C2×C42⋊C2, C23×C8, C22×C4○D4, C2×C23.24D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C4○D8 [×4], C23×C4, C22×D4 [×2], C23.24D4 [×4], C22×C22⋊C4, C2×C4○D8 [×2], C2×C23.24D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000004
0000130
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
0160000
1600000
0001600
001000
0000143
00001414
,
0160000
100000
0001600
0016000
0000143
000033

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
 

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L4M···4X8A···8P
order12···2222222224···444444···48···8
size11···1222244441···122224···42···2

56 irreducible representations

dim11111111222
type+++++++++
imageC1C2C2C2C2C2C2C4D4D4C4○D8
kernelC2×C23.24D4C2×D4⋊C4C2×Q8⋊C4C23.24D4C2×C42⋊C2C23×C8C22×C4○D4C2×C4○D4C22×C4C24C22
# reps1228111167116

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{24}D_4
 
% in TeX
 
G:=Group("C2xC2^3.24D4");
 
// GroupNames label
 
G:=SmallGroup(128,1624);
 
// by ID
 
G=gap.SmallGroup(128,1624);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
 
// generators/relations
 

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