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G = C24.75D4order 128 = 27

30th non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.75D4, (C2×C8).196D4, C4.114(C4×D4), C22⋊Q813C4, C4.91C22≀C2, C2.2(C8⋊D4), C2.2(C8.D4), (C22×C4).290D4, C23.774(C2×D4), C22.4Q1647C2, C22.79(C8⋊C22), C23.81(C22⋊C4), (C23×C4).259C22, (C22×C8).389C22, C23.7Q8.16C2, (C22×Q8).19C22, C22.120(C4⋊D4), (C22×C4).1370C23, C4.86(C22.D4), C22.68(C8.C22), (C22×M4(2)).21C2, C2.26(C23.36D4), C2.33(C23.23D4), C2.21(C23.38D4), C4⋊C4.73(C2×C4), (C2×Q8).70(C2×C4), (C2×C22⋊Q8).9C2, (C2×Q8⋊C4)⋊45C2, (C2×C4).1333(C2×D4), (C2×C4⋊C4).62C22, (C2×C4).567(C4○D4), (C22×C4).280(C2×C4), (C2×C4).388(C22×C4), (C2×C4).132(C22⋊C4), C22.269(C2×C22⋊C4), SmallGroup(128,626)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.75D4
C1C2C22C23C22×C4C23×C4C22×M4(2) — C24.75D4
C1C2C2×C4 — C24.75D4
C1C23C23×C4 — C24.75D4
C1C2C2C22×C4 — C24.75D4

Generators and relations for C24.75D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 356 in 180 conjugacy classes, 64 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×24], Q8 [×6], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42, Q8⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C22×Q8, C22.4Q16 [×2], C23.7Q8, C2×Q8⋊C4 [×2], C2×C22⋊Q8, C22×M4(2), C24.75D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C8⋊C22, C8.C22 [×3], C23.23D4, C23.36D4, C23.38D4, C8⋊D4 [×2], C8.D4 [×2], C24.75D4

Smallest permutation representation of C24.75D4
On 64 points
Generators in S64
(2 6)(4 8)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 24 5 20)(2 60 6 64)(3 22 7 18)(4 58 8 62)(9 55 13 51)(10 39 14 35)(11 53 15 49)(12 37 16 33)(17 47 21 43)(19 45 23 41)(25 34 29 38)(26 56 30 52)(27 40 31 36)(28 54 32 50)(42 63 46 59)(44 61 48 57)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim1111111222244
type++++++++++-
imageC1C2C2C2C2C2C4D4D4D4C4○D4C8⋊C22C8.C22
kernelC24.75D4C22.4Q16C23.7Q8C2×Q8⋊C4C2×C22⋊Q8C22×M4(2)C22⋊Q8C2×C8C22×C4C24C2×C4C22C22
# reps1212118431413

Matrix representation of C24.75D4 in GL8(𝔽17)

10000000
616000000
001600000
000160000
00001000
00000100
000000160
000000016
,
160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
10000000
616000000
00490000
004130000
00000010
00000001
00000100
000016000
,
112000000
86000000
006110000
003110000
00000064
000000411
0000111300
000013600

G:=sub<GL(8,GF(17))| [1,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[11,8,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,0,11,13,0,0,0,0,0,0,13,6,0,0,0,0,6,4,0,0,0,0,0,0,4,11,0,0] >;

C24.75D4 in GAP, Magma, Sage, TeX

C_2^4._{75}D_4
% in TeX

G:=Group("C2^4.75D4");
// GroupNames label

G:=SmallGroup(128,626);
// by ID

G=gap.SmallGroup(128,626);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,2019,248]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations

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