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G = C24.195C23  order 128 = 27

35th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.195C23, C23.204C24, C22.432+ (1+4), C22.272- (1+4), C22.43(C4×D4), (C22×C4).167D4, C22.D4⋊7C4, C23.607(C2×D4), C23.8Q8⋊6C2, C2.4(C23⋊3D4), C22.95(C23×C4), (C23×C4).45C22, C22.92(C22×D4), C23.224(C4○D4), C23.34D4⋊12C2, C23.125(C22×C4), (C22×C4).469C23, (C2×C42).411C22, C24.C22⋊5C2, C23.23D4.4C2, C23.63C23⋊4C2, (C22×D4).477C22, C23.65C23⋊12C2, C2.C42.40C22, C2.3(C22.33C24), C2.4(C23.38C23), C2.11(C23.33C23), C4⋊C4⋊9(C2×C4), C2.21(C2×C4×D4), (C2×C4×D4).31C2, (C22×C4⋊C4)⋊8C2, C22⋊C4⋊9(C2×C4), (C4×C22⋊C4)⋊32C2, (C22×C4)⋊23(C2×C4), (C2×C4).676(C2×D4), (C2×D4).167(C2×C4), (C2×C4).25(C22×C4), C22.89(C2×C4○D4), (C2×C4⋊C4).177C22, (C2×C22.D4).5C2, (C2×C22⋊C4).426C22, SmallGroup(128,1054)

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

Derived series
C1 — C22 — C24.195C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C24.195C23
Lower central
C1 — C22 — C24.195C23
Upper central
C1 — C23 — C24.195C23
Jennings
C1 — C23 — C24.195C23

Subgroups: 556 in 316 conjugacy classes, 148 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×20], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×14], C2×C4 [×52], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×4], C22⋊C4 [×12], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×5], C22×C4 [×16], C22×C4 [×14], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C4×D4 [×4], C22.D4 [×8], C23×C4 [×4], C22×D4, C4×C22⋊C4, C23.34D4, C23.8Q8, C23.8Q8 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C22×C4⋊C4, C2×C4×D4, C2×C22.D4, C24.195C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C2×C4×D4, C23.33C23 [×2], C23⋊3D4, C23.38C23, C22.33C24 [×2], C24.195C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, gfg-1=bcf >

Smallest permutation representation
►On 64 points
Generators in S64
(2 13)(4 15)(5 64)(6 38)(7 62)(8 40)(10 35)(12 33)(17 50)(19 52)(21 27)(22 57)(23 25)(24 59)(26 41)(28 43)(30 55)(32 53)(37 46)(39 48)(42 60)(44 58)(45 63)(47 61)

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
04000000
00100000
00040000
00001000
00000400
00000040
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
20000000
02000000
00100000
00010000
00000040
00000004
00004000
00000400
,
01000000
10000000
00010000
00100000
00000200
00002000
00000003
00000030
,
40000000
01000000
00400000
00010000
00000100
00004000
00000004
00000010

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4AD
order12···22222224···44···4
size11···12222442···24···4

44 irreducible representations

dim1111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4C4○D42+ (1+4)2- (1+4)
kernelC24.195C23C4×C22⋊C4C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C22×C4⋊C4C2×C4×D4C2×C22.D4C22.D4C22×C4C23C22C22
# reps11131222111164422

In GAP, Magma, Sage, TeX 

C_2^4._{195}C_2^3
 
% in TeX
 
G:=Group("C2^4.195C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1054);
 
// by ID
 
G=gap.SmallGroup(128,1054);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,100,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=b*c*f>;
 
// generators/relations
 

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