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G = C22.148C25  order 128 = 27

129th central stem extension by C22 of C25

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

Aliases: C23.89C24, C22.148C25, C42.585C23, C4.1142- (1+4), C22.242+ (1+4), D4⋊6D4⋊43C2, D4⋊3Q8⋊43C2, C4⋊C4.508C23, (C2×C4).138C24, C4⋊Q8.356C22, (C4×D4).256C22, (C2×D4).337C23, (C2×Q8).314C23, (C4×Q8).243C22, C4⋊D4.123C22, C4⋊1D4.194C22, C22⋊C4.118C23, (C22×C4).407C23, (C2×C42).975C22, C22⋊Q8.130C22, C2.73(C2×2+ (1+4)), C2.53(C2×2- (1+4)), C2.59(C2.C25), C22.26C24⋊55C2, C42⋊2C2.27C22, C4.4D4.183C22, C42.C2.168C22, C22.57C24⋊16C2, C22.34C24⋊26C2, C22.56C24⋊14C2, C22.31C24⋊28C2, C23.41C23⋊24C2, C22.47C24⋊37C2, C22.33C24⋊21C2, C23.36C23⋊57C2, C42⋊C2.251C22, C22.D4.21C22, (C2×C42.C2)⋊51C2, (C2×C4⋊C4).727C22, (C2×C4○D4).246C22, SmallGroup(128,2291)

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

Derived series
C1 — C22 — C22.148C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42.C2 — C22.148C25
Lower central
C1 — C22 — C22.148C25
Upper central
C1 — C22 — C22.148C25
Jennings
C1 — C22 — C22.148C25

Subgroups: 756 in 512 conjugacy classes, 382 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×23], C22, C22 [×2], C22 [×20], C2×C4 [×6], C2×C4 [×18], C2×C4 [×31], D4 [×29], Q8 [×7], C23, C23 [×6], C42 [×4], C42 [×6], C22⋊C4 [×38], C4⋊C4 [×2], C4⋊C4 [×52], C22×C4 [×3], C22×C4 [×22], C2×D4, C2×D4 [×20], C2×Q8, C2×Q8 [×6], C4○D4 [×12], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C42⋊C2 [×4], C4×D4, C4×D4 [×18], C4×Q8, C4⋊D4, C4⋊D4 [×24], C22⋊Q8, C22⋊Q8 [×24], C22.D4 [×26], C4.4D4, C4.4D4 [×2], C42.C2, C42.C2 [×14], C42⋊2C2 [×10], C4⋊1D4, C4⋊Q8, C4⋊Q8 [×6], C2×C4○D4 [×6], C2×C42.C2, C23.36C23, C22.26C24, C22.31C24 [×2], C22.33C24 [×6], C22.34C24 [×2], C23.41C23 [×2], D4⋊6D4 [×6], C22.47C24 [×4], D4⋊3Q8 [×2], C22.56C24 [×2], C22.57C24 [×2], C22.148C25

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, C22.148C25

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01300000
40020000
40010000
01400000
00001000
00000400
00000010
00001014
,
01300000
10030000
00040000
00400000
00000200
00002000
00003234
00003202
,
40000000
04000000
04100000
40010000
00003000
00000300
00000030
00000003
,
01000000
40000000
40010000
01400000
00000010
00001413
00001000
00000001
,
40000000
04000000
00400000
00040000
00000100
00001000
00001413
00000004

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4Z
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim1111111111111444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C22- (1+4)2+ (1+4)C2.C25
kernelC22.148C25C2×C42.C2C23.36C23C22.26C24C22.31C24C22.33C24C22.34C24C23.41C23D4⋊6D4C22.47C24D4⋊3Q8C22.56C24C22.57C24C4C22C2
# reps1111262264222222

In GAP, Magma, Sage, TeX 

C_2^2._{148}C_2^5
 
% in TeX
 
G:=Group("C2^2.148C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2291);
 
// by ID
 
G=gap.SmallGroup(128,2291);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,184,2019,570,248,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=a,f^2=b,a*b=b*a,d*c*d^-1=g*c*g=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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