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G = C22.156C25  order 128 = 27

137th central stem extension by C22 of C25

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

Aliases: C23.96C24, C22.156C25, C42.138C23, C4⋊C4.338C23, (C2×C4).146C24, (C4×D4).259C22, (C2×D4).344C23, (C2×Q8).321C23, (C4×Q8).246C22, C4⋊1D4.122C22, C4⋊D4.126C22, C22⋊C4.122C23, (C2×C42).980C22, (C22×C4).415C23, C22⋊Q8.132C22, C42.C2.90C22, C2.67(C2.C25), C42⋊2C2.28C22, C22.58C24⋊5C2, C4.4D4.185C22, C23.36C23⋊64C2, C22.46C24⋊43C2, C22.56C24⋊19C2, C22.47C24⋊42C2, C22.33C24⋊27C2, C42⋊C2.254C22, C22.34C24⋊29C2, C22.D4.40C22, (C2×C4⋊C4).730C22, SmallGroup(128,2299)

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

Derived series
C1 — C22 — C22.156C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.36C23 — C22.156C25
Lower central
C1 — C22 — C22.156C25
Upper central
C1 — C22 — C22.156C25
Jennings
C1 — C22 — C22.156C25

Subgroups: 676 in 475 conjugacy classes, 378 normal (7 characteristic)
C1, C2 [×3], C2 [×7], C4 [×24], C22, C22 [×21], C2×C4 [×24], C2×C4 [×21], D4 [×21], Q8 [×3], C23, C23 [×6], C42, C42 [×13], C22⋊C4 [×42], C4⋊C4 [×54], C22×C4 [×21], C2×D4 [×21], C2×Q8 [×3], C2×C42, C2×C4⋊C4 [×6], C42⋊C2 [×12], C4×D4 [×21], C4×Q8 [×3], C4⋊D4 [×27], C22⋊Q8 [×15], C22.D4 [×30], C4.4D4 [×3], C42.C2 [×21], C42⋊2C2 [×14], C4⋊1D4 [×2], C23.36C23 [×3], C22.33C24 [×6], C22.34C24 [×6], C22.46C24 [×6], C22.47C24 [×6], C22.56C24 [×3], C22.58C24, C22.156C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25, C2.C25 [×3], C22.156C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=a, f2=g2=ba=ab, dcd=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=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 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 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 57)(10 58)(11 59)(12 60)(21 34)(22 35)(23 36)(24 33)(25 30)(26 31)(27 32)(28 29)(37 51)(38 52)(39 49)(40 50)(41 46)(42 47)(43 48)(44 45)(53 61)(54 62)(55 63)(56 64)

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
00100000
00010000
10000000
01000000
00001040
00000401
00000040
00000001
,
10000000
04000000
00100000
00040000
00002030
00000203
00004030
00000403
,
01000000
10000000
00040000
00400000
00002000
00000200
00000020
00000002
,
20000000
02000000
00300000
00030000
00000400
00001000
00000301
00002040
,
30000000
03000000
00300000
00030000
00000100
00004000
00000001
00000040

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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,1,0,1],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,2,0,4,0,0,0,0,3,0,3,0,0,0,0,0,0,3,0,3],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,4,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

38 conjugacy classes

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

38 irreducible representations

dim111111114
type++++++++
imageC1C2C2C2C2C2C2C2C2.C25
kernelC22.156C25C23.36C23C22.33C24C22.34C24C22.46C24C22.47C24C22.56C24C22.58C24C2
# reps136666316

In GAP, Magma, Sage, TeX 

C_2^2._{156}C_2^5
 
% in TeX
 
G:=Group("C2^2.156C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2299);
 
// by ID
 
G=gap.SmallGroup(128,2299);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,360,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=a,f^2=g^2=b*a=a*b,d*c*d=g*c*g^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f^-1=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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