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G = C22.152C25  order 128 = 27

133rd central stem extension by C22 of C25

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

Aliases: C22.152C25, C42.134C23, C23.147C24, C4.1152- 1+4, C22.132- 1+4, D4⋊3Q8⋊44C2, Q8⋊3Q8⋊30C2, C4⋊C4.335C23, (C2×C4).142C24, C4⋊Q8.357C22, (C4×D4).257C22, (C2×D4).488C23, (C2×Q8).317C23, (C4×Q8).244C22, C4⋊D4.238C22, C22⋊C4.120C23, (C2×C42).977C22, (C22×C4).411C23, C22⋊Q8.131C22, C2.55(C2×2- 1+4), C42.C2.88C22, C2.63(C2.C25), C42⋊2C2.10C22, C22.58C24⋊4C2, C4.4D4.184C22, C23.37C23⋊56C2, C23.41C23⋊25C2, C22.57C24⋊19C2, C22.35C24⋊24C2, C22.46C24⋊39C2, C42⋊C2.252C22, C22.33C24.4C2, C22.D4.39C22, C23.36C23.34C2, (C2×C42.C2)⋊52C2, (C2×C4⋊C4).728C22, SmallGroup(128,2295)

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

Derived series
C1 — C22 — C22.152C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42.C2 — C22.152C25
Lower central
C1 — C22 — C22.152C25
Upper central
C1 — C22 — C22.152C25
Jennings
C1 — C22 — C22.152C25

Generators and relations for C22.152C25
 G = < a,b,c,d,e,f,g | a2=b2=g2=1, c2=f2=b, d2=e2=a, 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 >

Subgroups: 564 in 456 conjugacy classes, 382 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×C42.C2, C23.36C23, C23.37C23, C22.33C24, C22.35C24, C23.41C23, C22.46C24, D4⋊3Q8, Q8⋊3Q8, C22.57C24, C22.58C24, C22.152C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C2.C25, C22.152C25

Smallest permutation representation of C22.152C25
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 27)(6 28)(7 25)(8 26)(13 32)(14 29)(15 30)(16 31)(17 38)(18 39)(19 40)(20 37)(21 55)(22 56)(23 53)(24 54)(33 63)(34 64)(35 61)(36 62)(41 52)(42 49)(43 50)(44 51)(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 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 50 9 43)(2 44 10 51)(3 52 11 41)(4 42 12 49)(5 47 27 59)(6 60 28 48)(7 45 25 57)(8 58 26 46)(13 64 32 34)(14 35 29 61)(15 62 30 36)(16 33 31 63)(17 54 38 24)(18 21 39 55)(19 56 40 22)(20 23 37 53)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4AD
order1222222244444···4
size1111224422224···4

38 irreducible representations

dim111111111111444
type++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C22- 1+42- 1+4C2.C25
kernelC22.152C25C2×C42.C2C23.36C23C23.37C23C22.33C24C22.35C24C23.41C23C22.46C24D4⋊3Q8Q8⋊3Q8C22.57C24C22.58C24C4C22C2
# reps111126482222222

Matrix representation of C22.152C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
20000000
03000000
00300000
00020000
00002020
00004024
00000030
00003140
,
00100000
00010000
40000000
04000000
00001010
00000013
00000040
00000220
,
03000000
30000000
00020000
00200000
00002020
00000021
00001030
00003140
,
02000000
20000000
00030000
00300000
00003200
00000200
00000301
00000440
,
01000000
10000000
00010000
00100000
00004000
00000400
00000040
00000004

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

C22.152C25 in GAP, Magma, Sage, TeX 

C_2^2._{152}C_2^5
 
% in TeX
 
G:=Group("C2^2.152C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2295);
 
// by ID
 
G=gap.SmallGroup(128,2295);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,1430,723,184,2019,570,248,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=g^2=1,c^2=f^2=b,d^2=e^2=a,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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