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G = C22.144C25  order 128 = 27

125th central stem extension by C22 of C25

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

Aliases: C23.85C24, C22.144C25, C42.127C23, C4.472- (1+4), (Q82)⋊14C2, Q8⋊3Q8⋊27C2, D4⋊3Q8⋊41C2, C4⋊C4.329C23, (C2×C4).134C24, C4⋊Q8.355C22, (C4×D4).254C22, (C2×D4).334C23, C22⋊C4.58C23, (C4×Q8).240C22, (C2×Q8).470C23, C4⋊D4.237C22, (C22×C4).404C23, (C2×C42).972C22, C22⋊Q8.128C22, C2.50(C2×2- (1+4)), C42.C2.86C22, C2.56(C2.C25), C42⋊2C2.26C22, C4.4D4.107C22, C23.41C23⋊23C2, C22.35C24⋊21C2, C22.50C24⋊36C2, C22.46C24⋊37C2, C22.57C24⋊14C2, C42⋊C2.248C22, C23.37C23⋊53C2, C22.36C24.6C2, C22.D4.20C22, C23.36C23.33C2, (C2×C4⋊C4).725C22, SmallGroup(128,2287)

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

Derived series
C1 — C22 — C22.144C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.144C25
Lower central
C1 — C22 — C22.144C25
Upper central
C1 — C22 — C22.144C25
Jennings
C1 — C22 — C22.144C25

Subgroups: 564 in 455 conjugacy classes, 382 normal (50 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×26], C22, C22 [×9], C2×C4 [×6], C2×C4 [×22], C2×C4 [×13], D4 [×4], Q8 [×18], C23, C23 [×2], C42 [×4], C42 [×22], C22⋊C4 [×30], C4⋊C4 [×4], C4⋊C4 [×70], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×10], C2×C42, C2×C4⋊C4 [×2], C42⋊C2 [×12], C4×D4, C4×D4 [×4], C4×Q8 [×3], C4×Q8 [×20], C4⋊D4, C22⋊Q8 [×3], C22⋊Q8 [×22], C22.D4 [×6], C4.4D4, C4.4D4 [×6], C42.C2, C42.C2 [×28], C42⋊2C2 [×26], C4⋊Q8 [×4], C4⋊Q8 [×14], C23.36C23, C23.37C23 [×2], C22.35C24 [×8], C22.36C24 [×2], C23.41C23 [×2], C22.46C24 [×2], D4⋊3Q8, C22.50C24, C22.50C24 [×4], Q8⋊3Q8, Q8⋊3Q8 [×2], Q82, C22.57C24 [×4], C22.144C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2- (1+4) [×4], C25, C2×2- (1+4) [×2], C2.C25, C22.144C25

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

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

(2 52)(4 50)(5 7)(6 35)(8 33)(10 54)(12 56)(13 15)(14 60)(16 58)(17 19)(18 64)(20 62)(22 38)(24 40)(26 42)(28 44)(29 31)(30 48)(32 46)(34 36)(45 47)(57 59)(61 63)

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40300000
42030000
00100000
24130000
00000020
00004424
00002000
00000211
,
30000000
42000000
00300000
30420000
00000010
00003343
00004000
00000002
,
22000000
03000000
21330000
00020000
00002000
00000200
00000030
00001103
,
10000000
01000000
40400000
42040000
00001000
00000100
00000040
00003304
,
10000000
01000000
00100000
00010000
00000100
00004000
00003343
00002011

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G···4AE
order12222224···44···4
size11114442···24···4

38 irreducible representations

dim11111111111144
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C22- (1+4)C2.C25
kernelC22.144C25C23.36C23C23.37C23C22.35C24C22.36C24C23.41C23C22.46C24D4⋊3Q8C22.50C24Q8⋊3Q8Q82C22.57C24C4C2
# reps11282221531442

In GAP, Magma, Sage, TeX 

C_2^2._{144}C_2^5
 
% in TeX
 
G:=Group("C2^2.144C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2287);
 
// by ID
 
G=gap.SmallGroup(128,2287);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,723,352,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=f^2=1,c^2=g^2=a,d^2=e^2=b,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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