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G = C24.202C23  order 128 = 27

42nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.202C23, C23.215C24, C22.532+ (1+4), C4⋊D4⋊18C4, C23.23D4⋊9C2, C23.14(C22×C4), (C2×C42).14C22, C23.34D4⋊13C2, C22.106(C23×C4), (C23×C4).294C22, (C22×C4).480C23, C24.C22⋊9C2, C24.3C22⋊16C2, C23.63C23⋊9C2, C2.6(C22.32C24), (C22×D4).107C22, C2.20(C22.11C24), C2.C42.470C22, C2.6(C22.34C24), C4⋊C4⋊12(C2×C4), (C2×D4)⋊17(C2×C4), C2.18(C4×C4○D4), C22⋊C4⋊28(C2×C4), (C4×C22⋊C4)⋊33C2, (C22×C4)⋊26(C2×C4), (C2×C4⋊D4).17C2, (C2×C4).35(C22×C4), (C2×C4).517(C4○D4), (C2×C4⋊C4).184C22, C22.100(C2×C4○D4), (C2×C22⋊C4).430C22, SmallGroup(128,1065)

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

Derived series
C1 — C22 — C24.202C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.202C23
Lower central
C1 — C22 — C24.202C23
Upper central
C1 — C23 — C24.202C23
Jennings
C1 — C23 — C24.202C23

Subgroups: 604 in 298 conjugacy classes, 136 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×18], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×12], C2×C4 [×42], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×14], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×14], C22×C4 [×10], C2×D4 [×12], C2×D4 [×6], C24, C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4, C4×C22⋊C4 [×2], C23.34D4, C23.23D4 [×4], C23.63C23 [×2], C24.C22 [×2], C24.3C22 [×2], C2×C4⋊D4, C24.202C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], 2+ (1+4) [×4], C4×C4○D4, C22.11C24 [×2], C22.32C24 [×2], C22.34C24 [×2], C24.202C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=db=bd, g2=c, faf-1=ab=ba, eae-1=ac=ca, ad=da, ag=ga, bc=cb, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

01000000
10000000
00010000
00100000
00004000
00000400
00001010
00000101
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
02000000
30000000
00040000
00100000
00000403
00004030
00000101
00001010
,
10000000
01000000
00300000
00030000
00004030
00000403
00000010
00000001
,
20000000
02000000
00200000
00020000
00000100
00004000
00000001
00000040

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

44 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4AD
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim11111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4○D42+ (1+4)
kernelC24.202C23C4×C22⋊C4C23.34D4C23.23D4C23.63C23C24.C22C24.3C22C2×C4⋊D4C4⋊D4C2×C4C22
# reps131422211684

In GAP, Magma, Sage, TeX 

C_2^4._{202}C_2^3
 
% in TeX
 
G:=Group("C2^4.202C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1065);
 
// by ID
 
G=gap.SmallGroup(128,1065);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,219,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=d*b=b*d,g^2=c,f*a*f^-1=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,f*e*f^-1=g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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