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G = C22.143C25  order 128 = 27

124th central stem extension by C22 of C25

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

Aliases: C23.84C24, C22.143C25, C42.126C23, C4.462- (1+4), C4.1622+ (1+4), (D4×Q8)⋊31C2, D4⋊6D4⋊42C2, Q8⋊6D4⋊30C2, D4⋊3Q8⋊40C2, Q8⋊5D4⋊32C2, Q8⋊3Q8⋊26C2, C4⋊C4.328C23, (C2×C4).133C24, C4⋊Q8.354C22, (C4×D4).253C22, (C2×D4).487C23, C22⋊C4.57C23, (C2×Q8).469C23, (C4×Q8).239C22, C4⋊D4.121C22, C4⋊1D4.193C22, (C2×C42).971C22, (C22×C4).403C23, C22⋊Q8.127C22, C2.71(C2×2+ (1+4)), C2.49(C2×2- (1+4)), C42.C2.85C22, C2.55(C2.C25), C42⋊2C2.25C22, C22.26C24⋊53C2, C4.4D4.106C22, (C22×Q8).373C22, C22.56C24⋊12C2, C23.36C23⋊54C2, C22.53C24⋊23C2, C23.38C23⋊36C2, C22.34C24⋊25C2, C22.33C24⋊19C2, C22.36C24⋊36C2, C42⋊C2.247C22, C23.37C23⋊52C2, C22.D4.19C22, (C2×C4⋊C4).724C22, (C2×C4○D4).245C22, SmallGroup(128,2286)

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

Derived series
C1 — C22 — C22.143C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.143C25
Lower central
C1 — C22 — C22.143C25
Upper central
C1 — C22 — C22.143C25
Jennings
C1 — C22 — C22.143C25

Subgroups: 756 in 511 conjugacy classes, 382 normal (50 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×22], C22, C22 [×21], C2×C4 [×6], C2×C4 [×18], C2×C4 [×25], D4 [×26], Q8 [×16], C23, C23 [×6], C42 [×4], C42 [×10], C22⋊C4 [×42], C4⋊C4 [×4], C4⋊C4 [×42], C22×C4 [×3], C22×C4 [×18], C2×D4, C2×D4 [×20], C2×Q8 [×3], C2×Q8 [×8], C2×Q8 [×4], C4○D4 [×8], C2×C42, C2×C4⋊C4 [×4], C42⋊C2 [×4], C4×D4, C4×D4 [×18], C4×Q8 [×3], C4×Q8 [×6], C4⋊D4, C4⋊D4 [×18], C22⋊Q8 [×3], C22⋊Q8 [×24], C22.D4 [×26], C4.4D4, C4.4D4 [×14], C42.C2, C42.C2 [×8], C42⋊2C2 [×6], C4⋊1D4, C4⋊1D4 [×2], C4⋊Q8 [×3], C4⋊Q8 [×4], C22×Q8 [×2], C2×C4○D4 [×4], C23.36C23, C22.26C24, C23.37C23, C23.38C23 [×2], C22.33C24 [×4], C22.34C24 [×2], C22.36C24 [×4], D4⋊6D4 [×2], Q8⋊5D4 [×2], D4×Q8, Q8⋊6D4, D4⋊3Q8 [×2], Q8⋊3Q8, C22.53C24, C22.53C24 [×2], C22.56C24 [×4], C22.143C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×2], 2- (1+4) [×2], C25, C2×2+ (1+4), C2×2- (1+4), C2.C25, C22.143C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=b, g2=a, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
02000000
30000000
00020000
00300000
00004000
00003100
00000010
00001444
,
40300000
04030000
10100000
01010000
00000010
00000403
00004000
00000101
,
03000000
30000000
02020000
20200000
00004100
00003100
00000112
00001444
,
01000000
10000000
00010000
00100000
00002300
00004300
00000331
00003222
,
40000000
04000000
00400000
00040000
00004100
00003100
00000443
00001011

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

38 conjugacy classes

class 1 2A2B2C2D···2J4A···4F4G···4AA
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111111111111111444
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2- (1+4)C2.C25
kernelC22.143C25C23.36C23C22.26C24C23.37C23C23.38C23C22.33C24C22.34C24C22.36C24D4⋊6D4Q8⋊5D4D4×Q8Q8⋊6D4D4⋊3Q8Q8⋊3Q8C22.53C24C22.56C24C4C4C2
# reps1111242422112134222

In GAP, Magma, Sage, TeX 

C_2^2._{143}C_2^5
 
% in TeX
 
G:=Group("C2^2.143C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2286);
 
// by ID
 
G=gap.SmallGroup(128,2286);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,352,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b,g^2=a,a*b=b*a,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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